In short no. Historians and biblical scholars generally consider the only material which can provide an insight into the historical Jesus to date from the 1st and 2nd centuries CE. Specially the three synoptic Gospels (Mark, Matthew and Luke), the Gospel of John, the letters of Paul which mention biographical details of Jesus (primarily Romans and 1 Corinthians), as well as brief mentions by the Jewish historian Josephus and the the Roman historians Suetonius and Tacitus. Many scholars also believe in the existence of a lost source named ‘Q’ (from Quelle: source) which alongside the Gospel of Mark was the major source for Matthew and Luke. Within these sources there’s a huge amount of disagreement about what’s authentic. For example Josephus’ text has obviously been altered by later writers to fit with Christian theology, while the Gospel of John is a much later text than the other three gospels and contradicts is predecessors repeatedly.
There are several texts which date from the late 2nd-4th centuries which claim to provide additional biographical details about Jesus. Some of these are embellishments to the tradition by authors claiming to be biblical figures, for example the infancy gospel of Thomas (written circa 150 CE) gives us our image of an ox and an as at the nativity. The infancy Gospel also adds tales of Jesus’ childhood, informing its readers how a young Jesus matured and learned to control his powers (he murders several other children and withers his teacher’s hand). More benignly he also brings to life birds made of clay.
Other texts from the this period have more obvious theological agendas and seek to present a Jesus more in line with their churches’ theologies. The Gospels of Judas, Mary Magdalene and Thomas for example are all written from a Gnostic point of view, and present Jesus as a purely spiritual being, sent to liberate humanity. The Gospel of Judas in particular rewrites the narrative making Judas the hero, who acted on Jesus’ orders to free his master.
Other Gnostic texts from this time attempt to address the issue of Jesus as a god executed like a common criminal. If Jesus was a non material being, why did he appear to suffer? The solution put forward by some authors is the theory of substitution, the notion that Jesus traded places with another person who died on the cross. The brilliantly titled ‘Second Treatise of the Great Seth’ skirts around problems of authority by claiming to be a first person account by Jesus. In it Jesus switches places with Simon of Cyrene who dies while Jesus spiritually ascends into heaven. In the Gospel of Barnabas the same process occurs, but with Judas taking Jesus’ place.
According to Islamic tradition the Quran was completed in 632 CE, meaning that its distance from the historical Jesus is the same as our historical distance from the reign of Tamerlane making it unlikely to impossible that it contains any accurate ‘new’ information. Rather the author is keen to stress that he is not saying anything original about Jesus. Jesus mentioned in 93 Quranic verses, some of which echo the non canonical gospels, for example the story of the clay birds is repeated in Q 3:49, while the notion of the substituted Jesus appears in Q 4: 157-158. Crucially although the Quranic text is drawing from non biblical and possibly gnostic sources about Jesus, it has a very different agenda. In the 93 Jesus verses, the majority extol his prophethood and in the words of Greg Barker “point to the sovereignty of God she the truth of the message later transmitted by Mohammed’. A good example of this is 3:49 which makes it very clear that Jesus’ power only comes through the authority of Allah “with God’s permission it will become a real bird’. Although the Quran is utilising these stories, ultimately it is repurposing them to try and demonstrate Jesus is not part of the Christian Holy Trinity, or a Gnostic Spiritual redeemer. Instead it’s purpose is to show that his life and teachings form part of a chain of Prophets beginning with Adam, and ending with Muhammad who preached a consistent message. To quote the book itself “we believe in what was sent down to us and what was sent down to Abraham, Ishmael, Issac, Jacob and the Tribes, and what was given to Moses, Jesus and all the Prophets by their Lord. We make no distinction between any of them’.
Ultimately the Quran tells us nothing new about the historical Jesus, but tells us a great deal about conflicts over the identity of Jesus in the centuries following his death.
Edit: Recommended Sources
Jesus Tradition in Paul’s Letters by David Capes
Jesus in the World’s Faiths by Gregory Barker
Jesus in the World’s Religions: The Classic Texts e.d by Barker and Gregg
The Nag Hammadi Scriptures e.d by Marvin Meyer
‘The Gnostic Gospels’ and ‘The Gospel of Judas’ both by Elaine Pagels
Nag Hammadi, Gnosticism and Early Christianity e.d by Hedrick and Hodgeson
Jesus in the Nag Hammadi Writings by Majella Franzmann
The Islamic Jesus by Mustafa Akyol
Jesus in the Quran by Geoffrey Parrinder
I have to disagree with /u/NumisAl, who has nonetheless written an excellent response, particularly regarding apocrypha like the Gospel of Thomas. From a scholarly standpoint, however—and this is not necessarily a reflection of my personal beliefs—I am somewhat less confident in the gospels and I have a few reasons to be more optimistic in the Qu'ran than /u/NumisAl suggests.
As a historian, I often approach questions of religion cautiously, particularly when addressing questions about what "actually" happened. These kinds of questions used to be the primary subject matter for historians. Leopold von Ranke (1795-1896) famously claimed that the purpose of the historical writing should be to show how things really were (wie es eigentlich gewesen).
Attitudes have changed over time, and historians now have much more awareness about the things we just can't know. Following von Ranke's dictum, many historians started making assumptions, filling in gaps with common sense inferences, and these kinds of things. But as archives became better organized, other kinds of evidence like archaeology gained attention, and more international collaboration became possible, we learned that trying to show the past as it actually was could often be an impractical goal. Instead, we turned toward more practical questions of what can our sources actually show us.
I'm not sure if there's an easy quote to show the difference of this perspective, but Jacques Derrida's famous statement—There is nothing outside the text (il n'y a pas de hors-texte)—comes to mind. Most historians nonetheless reject a literal understanding of Derrida's axiom. We tend to think that our sources do represent historical truth, but we need to be honest about just how much (or how little) our sources can actually reveal.
Turning toward the specifics of your question, then, the basic problem is that these are very different kinds of sources. The gospels reflect ideas about Jesus that were recorded within about 70 years of his death. The Qur'an reflects ideas about Jesus that people were ready to accept about 600 years after his death. This difference in time immediately suggests that more reliable information might be found in the gospels, but we also need to consider how stories were preserved in each society.
In the Hellenistic world that the gospels came out of, we get a sense that people writing stories about the past rarely considered literal truth to be the most important priority. Typically, they thought history was meant to have a moral purpose, and so they would manipulate their materials to communicate this. Our clearest example is how the authors of Matthew and Luke grafted quotations from the Q source onto the base narrative by Mark. Matthew and Luke (not necessarily the actual names of the authors, but this is how we conventionally refer to them) inserted these quotes at different points in their narratives. This means we can probably doubt the literal truth of the way they describe surrounding events, but nonetheless both gospels end up pointing in different ways to a single common story, although spun differently for different audiences. From this perspective, the literal truth can sometimes be sacrificed to ensure that deeper truths are communicated effectively. (Does it really matter when Jesus said something, or if he actually cobbled all of his greatest hits into a Sermon on the Mount?)
In the late-antique Arabic world that the Qur'an came out of, there seem to have been rich traditions of story telling. (I'm not sure if many of the written apocrypha would have circulated among these societies, which is part of why I'm not sure if critiques about the Egyptian Gospel of Thomas and other gnostic texts should be applied to Arabia.) Early Arabic stories were crafted for intelligent and discerning audiences, since we know that early Arabic poets could come up with some beautiful and complex things. But there also seems to have been a deep commitment toward literal truth and citation, even though memories were preserved in oral form. We see this most clearly in the hadith, or the stories about the Prophet Muhammad, which were often passed down for centuries before being written down. The early compilers were often able to catalog many different versions of the same story, typically with only minor differences, and they could trace these stories from person to person back to the original eye witnesses. It seems reasonable to assume that these practices didn't come out of nowhere, and pre-Islamic Arabic culture might have had a similarly complex dedication to the preservation of historical truth. Arguably, the multi-ethnic communities of late-antique Arabia might have had a better shot at preserving historical truths than the theologically- and politically-charged environment of the Roman Near East.
Of course, we also don't have original copies of these texts, so we also need to think about how they were preserved. Here too they were preserved. We know that early Christians didn't necessarily consider the gospels inviolable scriptures. Scribes occasionally made "corrections," adding explanatory phrases, inserting stories the original authors had left out, or simply changing things that they considered errors. In most cases, we can figure out what these additions were, since they only occur in a few copies, but sometimes it becomes harder to know. With the Gospel of Mark, for example, we don't even know if it was supposed to have a longer ending.
The Qur'an seems to have been treated more like an unchangeable scripture from the outset, although here too there was room for interpretation and change. There was probably less outright intervention than we see for the Christian gospels, but early Arabic typically wasn't written with dots, which meant you often needed to know what a text said before you could read it. Often time, context helped, just like we can usually see without thinking if we're looking at a capital I or a lower-case L, but there's much more room for variety in Arabic. For example, a short vertical line could represent b, n, t, or th. That's in addition to the fact that short vowels were typically not written. By the 900s, Islamic scholars had worked out seven canonical ways to read the quranic text by inserting different vowels and dots. Unfortunately, much of the scholarly criticism that surrounded this diversity of readings has been forgotten since printed copies of the Quran became popular in the early 1900s. A printed book tends to give the impression that there's only one final form of a text, and the version of the Qur'an that circulates most commonly today is a version that Egyptian officials selected in 1924 (although it goes back to at least the 700s). Unfortunately, we still don't have a critical edition of the Qur'an that examines all the variants, placing them side by side, and cross-referencing the ancient manuscripts that preserve these readings. Until we do, it is difficult to state with certainty how divergent the quranic versions are, although it seems like differences are mostly minor, probably parallel to the same degrees of difference we typically see among early copies of the gospels.
TLDR We can't place the gospels against the Qur'an on a scale and weigh them in terms of historical accuracy. Both reflect different things. The gospels are closer in time, but Hellenistic writers showed a cavalier attitude (at least from a modern perspective) toward literal truth, and early scribes inserted additions and "corrections" that changed these texts, since they didn't think about them as being canonical scriptures for about 200 years. The Qur'an is much later, but it comes out of a society that appears to have held a deeper connection to literal truths and which had developed traditions that could preserve oral knowledge for hundreds of years with minimal changes. It's hard to imagine that the Qur'an could have attracted believers if its stories about Jesus didn't at least fall within the realm of the believable to contemporary Near Eastern audiences. Like most things with religion, it is, ultimately, a matter of faith.
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Two scholarly starting points:
Two popular reads that I'd recommend first (even for academics):