As a peasant in 14th century England, there is absolutely zero chance you would have a Quran. There is a teeny, tiny, possible chance you would know it existed.

There are several factors rendering this impossible: the expense of books, literacy and education, Latin Europe's knowledge of the Quran, and Latin Christianity's use of Islam.

First, books are a luxury item in the Middle Ages. The introduction of paper from the Arab world in the late Middle Ages starts to make pamphlets and broadsheets (somewhat more) affordable for urban middle classes in C15, but in C14 you are still talking parchment--animal skin, enormously wasteful and thus expensive. And to have enough for an entire Quran, not just a scrap of paper certifying an indulgence? Absolutely not going to happen in a peasant household, even if you (rightly) take "peasant" expansively to include petty freeholders (i.e. land) with enough money to hire a domestic servant or two.

Second, literacy. Any idea of literacy rates in the Middle Ages are wild guesstimates. This is complicated by the multilingual nature of texts and reading in the late Middle Ages--especially since we know increasing numbers of people (especially women) were reading in the vernacular, and in the Middle Ages literatus meant specifically Latin. Still, by 1500 we can posit literacy rates of 30-50% in cities, probably 10-15% for the population overall, and low single digits for rural peasantry. And that's in 1500, at the end of a century that saw an EXPLOSION in education, literacy, and book production. (The printing press cannot be profitable to invent or adopt unless there is already a waiting, eager audience to read the books it prints). In the 1300s? Peasants could probably recognize their names to scratch an X of "signature" on a contract or other legal document. And that's in vernacular English.

The Quran is in Arabic; it is a religious principle of Islam that the text not be translated. There is some fudging of this, especially in the Middle Ages with respect specifically to Persian. The vast, vast, vast majority of medieval Europeans had absolutely no reason or opportunity to learn to speak Arabic, much less read. In England, basically not at all.

Italian merchant families who had outposts in the Arab-speaking world (Cairo, Alexandria, Tunisia) probably had a few members, generally the ones resident abroad, who learned basic trade Arabic for economic purchases. (Although already there is a difference between spoken Arabic dialects and classical Quranic Arabic). Actual Muslims living in Latin Europe, by 1300, are basically limited to ever-shrinking areas of Iberia. Maybe some Spanish people learned a little Arabic, but we know the mudejar communities also functioned well in Romance.

A bare handful of studia arabica or Arabic schools existed, but they were extremely rare and even more extremely limited in student body and purpose: basically, training friars who would need some Arabic to minister to Christian (mostly merchant) populations in Islamic territory. And these schools were definitely not found in England, which is still more of a global backwater in the Mediterranean-centric 14th century.

Now, there were a handful of attempts to translate the Quran into Latin in the Middle Ages, for purposes of helping Europeans write anti-Muslim polemic. The most-copied version, however, only survives in 25 manuscripts (although certainly more were made that don't survive)--and we are still talking Latin, which medieval peasants did not learn to read. And at 100 folios or sheets of parchment, they categorically could not afford a copy.

And finally, medieval English peasants would almost certainly not have known the Quran existed, because of the way Islam and Muslims functioned or were used in Latin Christendom. (With the possible exception of some people in some Spanish cities with a mudejar community, who might have a vague idea of what "those neighbors" believed. We know Christians, Muslims, and Jews were ALL very good at manipulating the overlapping legal jurisdictions of Christian, Islamic, and Jewish courts in applicable situations to angle for personal advantage, so there is some idea of at least different foundational texts behind the laws there. But this is definitely not rural English peasantry).

There were a couple possible ways English peasants might have learned some VERY HEAVILY FILTERED AND ALTERED ideas about Islam and Muslims: popular stories and sermons. While illiterate peasants weren't reading romance stories (which might feature a "Moorish princess" or a "Saracen knight" who ultimately converts to Christianity) firsthand, there is evidence from the late 14C that some peasants were very well aware of popular English written texts through various means, so I wouldn't rule out knowledge of the romance tradition, either). And sermons, although surviving in Latin written copies, would have been preached in the vernacular. Medieval sermons drew heavily on "exempla" or sample anecdotes that the preacher would expand on to teach a lesson.

From the romance and other literary traditions, especially harkening back to the memory of the Crusades, medieval peasants would have had some idea of Islam as paganism, or rather of Saracens as pagans or sometimes heretics. (In the Song of Roland, for example, Muslims worship an anti-trinity of Mahomet, Apollo, and a god known as Termagant.)

Scholars have typically been more interested in how sermon exempla portray Jews rather than Muslims (owing to prevalence and also the role that exempla have been shown to play in reflecting/promoting the rising tide of anti-Judaism in the late Middle Ages). But there is some research that shows medieval preachers used Muslims as twisted mirrors of Christians--using Islamic immorality (as defined in a medieval Christian perspective, of course) to illuminate proper Christian action.

For example, to teach audiences that greed is bad: a Saracen convert to Christianity is described distributing alms to the poor, while his still-Muslim sons fight each other to the death over the inheritance that none of them are actually going to get. The central lesson here is about the sin of greed, but the vehicle also communicates the idea of Muslims/Islam as greedy.

There does seem to have been an idea among highly educated Christian clerics, especially those with firsthand exposure to Islam and Muslims like Jacques de Vitry (who traveled to the Near East and wrote a lot about it. A LOT about it), that Muslims would see the light pretty much automatically when exposed to the differences between the Quran and the Bible.

Whether this trope actually made it into preached sermons is impossible to say, though. Especially given the massively conflicting information about the education level of the average rural parish priest. And that's too bad, because that's the main way I can see medieval peasants in England even knowing that the Quran existed as a book.

So I guess there is a very, very, very outside chance that medieval English peasants in the 14th century would have known that the Quran existed. There is zero chance they would have been able to either own one or read one.