Well, to begin with, early christianity was very quick trying to found itself on written sources and testimonies (of varying accuracy and sourcing quality) about the life of Jesus. The earliest manuscript we have of the Gospel of John, the Bodmer Papyri, can be dated to around 150AD by the earliest estimates^1 and John was probably one of those composed the latest (as can be devised for example by its highly allegorical content with numerous references to the rest of the Jewish corpus, suggesting the emergence of a Christian theology, beyond a simple account of the life of Christ). It is also generally considered that Mark is the book that was composed the earliest, usually dated around 70AD^2. There are indications that most Gospels either were composed by eyewitnesses of the last years of Christ's life and teaching, or people who had known such eyewitnesses (in 66-70AD as Perkins dates Mark, there would have been such elderly eyewitnesses in early Christian communities). There are also indications of a far broader profusion of writings, of which we have kept some apocryphal gospels and books of revelations, between 50AD and 200AD (Gospel of Thomas, Gospel of Judas, Gospel of Barnabas) with a wide spectrum of relatively orthodox (from a later point of view) writings to gnostic heterodox writings. In any case, early Christianity was from the onset based on written accounts of the life of Christ and his teachings, and developped in preexisting cultures that put an extremely high emphasis on written teachings, both the Jewish culture of the time, which already placed a high importance on the written world of God, and the Hellenistic culture, which, while holding specific deference for spoken rational discourse (logos) had an extremely refined scholarly tradition. As such, later Christians, continuing in the Middle Ages, considered Scripture as, well, something written (rather than something memorized or said) -- Scripture comes from the Latin scribo, is, ere, ipsi, iptum, to write, to note in written form. The Good Book, is also, importantly, called the Bible (from the greek biblion, a book in written form).
On the contrary, early Islam did not emerge in a culture with such an emphasis on written teaching and scholarly comment but in a culture where a significant part of the population was nomadic, illiterate, and where most cultural transmission came orally. It is particularly telling that القرآن al-Qurʾān litterally means "the recitation" (and not the book, or anything meaning that it is supposed to be written). An important meaning of this word is in particular the "act of reciting", as reflected in an early Quranic passage: "It is for Us to collect it and to recite it (qurʼānahu)." (Quran, 75:17). There are, nonetheless indications that written depictions of Muhammad's teachings were kept quite early on : tradition holds it that the first of them was composed on orders of Abu Bakr, somewhere around 650AD (within 20 years of Muhammad's death in 632), and palimpsests were found in Yemen almost as early as Muhammad^3. There is also broad consensus that around these very first early written accounts (that would become the core of the Quran) most transmission went on orally, by people who had memorized parts of the early islamic teachings and sayings of Muhammad and his companions, and kept teachings of wisdom that were (much later) compiled as Hadith. In any case, there was a greater space for oral transmission in this context : islamic tradition considers that the oral memorization by hundreds of followers was essential to retaining perfect fidelity to the revelation as given by God to Muhammad.
Also do bear in mind that the Quran is composed as something relatively poetic or at least rythmic in nature, with the idea that it should be cantillated (not really sung per se, but a pattern of intonation that allows easier memorization). The gospels absolutely do not have this specificity, and would be much harder to memorize. From a purely pedagogical standground, medieval clergy did not rely on musical / rythmical methods to memorize religious texts but gave significant emphasis to spatial mnemotechniques first mentionned by Cicero in the De Oratore (the famous mind palace popularized by Sherlock), often using the depictions of the Temple of Jerusalem or of the Celestial Jerusalem in Revelations as the structure of their internal memories (hence why this spatial organization would be everywhere in monasteries)^4.
In conclusion : Christianity does not put the same emphasis on rote memorization and recitation of holy texts, and had, much earlier, a strong emphasis on written form and commentary. The Christian Holy Books were not composed to that end, and do not have the rythmic nature which allows for memorization of islamic teachings. To add on this, Medieval Christianity did not emphasize memorization based on melodies and/or rythmic patterns as Hafiz learning would, but relied much more on spatial memorization techniques.
^1 : Herbert Hunger. "Zur Datierung des Papyrus Bodmer II (P66)," Anzeiger der Österreichischen Akademie der Wissenschaften philosophisch-historische Klasse 97 (1961)
^2 : Perkins, Pheme (1998). "The Synoptic Gospels and the Acts of the Apostles: Telling the Christian Story". In Barton, John. The Cambridge companion to biblical interpretation. Westminster John Knox Press. pp. 241–58
^3 : Bergmann, Uwe; Sadeghi, Behnam (September 2010). "The Codex of a Companion of the Prophet and the Qurān of the Prophet". Arabica. 57 (4): 343–436.
^4 : The Temple of Jerusalem: From Moses to the Messiah, S. Fine
I cannot say whether christian's specifically knew about this practice. However, given the contact between Christians and Muslims in Spain, southern Italy and the Levant, even if Christendom as a whole did not know about the concept of a Hafiz, there would most likely have been Christian individuals who did. What I can say is that as a whole people of the past placed a higher premium on memory than we do today. this makes sense given that while written books and records did exist, they were much more expensive and much rarer. For example, if you’re a monk in Normandy and you’re looking for a specific book that’s only found in Rome you’ll probably have to march on down to Rome in order to read it. Once you get back, you’d better remember what you read otherwise you’ll have to march back down to Rome and read it again. Sure you could take notes or just copy the book. But your notes would never cover the whole text adequately and manuscript copying was incredibly expensive and laborious. Given the importance of memory, people in the middle ages utilized a number of techniques to aid in the memorization process.
A key difference between Christianity and Islam and memorizing their respective holy books is that the Quran was written down much quicker than the Bible was, but during its inception it was primarily meant to be memorized. The Quran was not written down until after Muhammad’s death, meaning that the text was primarily recorded in people’s memory and a special emphasis was placed upon memorizing the text. Christianity, on the other hand, did not have this early emphasis on memorization, and it took hundreds of years before the Christian community even decided on what books should be in the Bible. That is not to say that there were not Christians who did memorize the Bible, just that the tradition does not place a premium on it. For Christians specifically, memorizing the book of Psalms was the benchmark for literacy, and was even before the birth of Islam. The Life of St. Theodore of Sykeon mentions that at a young age Theodore attempted to learn the Psalter by heart, and ran into particular trouble with the 17th.
I'd be interested in hearing a more general answer than this, namely, is there evidence of *any* transfer of religious practices or beliefs from the Islamic to the Christian world during the medieval period.
I wanted to clarify that the meaning word Hafiz (Keeper/Guardian/Care Taker) has changed, the original meaning of the word Hafiz is someone who has fully memorised at least 20,000 Hadith including their chain of narration including ability to deduce the ranking of the hadith from the chain of narration. [Source Al-Nukat - Al-Zarkashi Part1/Page53]
Hence Ibn Hajar the famous prolific commentator on hadith is called a Hafiz. Nowadays it's incorrectly used for someone who has memorised the whole Quran.