It would be exceedingly unlikely for you to obtain a copy. Or really of any book.

Think about it this way: the Bible plays a much bigger role in your life - or at least parochial interpretations of the Bible, since you've never read it - and there is a likelihood that your own church doesn't even have one, just a priest or someone along those lines who has read it and is capable of interpreting it to a mostly or uniformly illiterate, lay diocese. The printing press doesn't really factor into European history until the 1430s-1440s, so you don't have many copies of the Bible out in circulation. And the ones you do have aren't written in the local vernacular. And they're really, really expensive because everything's still all done by hand.

I would imagine that many of the logistical reasons for why someone in the lower classes wouldn't have a Bible would be magnified (once you factor in geography and cultural differences) for why that same person wouldn't have a copy of the Qur'an.

Disregarding all of this, if you did have a copy of it, it would still be in Arabic and it's unlikely that you would run into a learned enough scholar who would know what it was. I don't really think it would be all that incriminating. I'm not an expert on this, but I don't think England in the 1300s really went through intense witch-hunts. I know that they did in the 1640s and that they passed a series of Witchcraft Acts starting in the 16th century, but you're a couple centuries away from that. I'm basing this speculation on what I know of the historical patterns of the time, but it's still speculation....But I don't think you'll have to worry about superstitious or vindictive neighbors ratting you out to get you burned at the stake.

Europeans on the continent, especially on the Iberian peninsula, would stand a much better chance of knowing what a Qur'an was due to Moorish influences on the culture there.

All this being said, I don't want to seem like I'm actually downplaying Islamic contributions to our ideas of Europe. Europe and Islam don't exist in completely different worlds; the argument could easily be made that if it wasn't for Islamic scholars, for instance, Aristotle's works would be lost to time.

This is another digression - I figure since you asked a question like this you might be curious about other interesting connections between England, or the British Isles in general, and Islam - but King Offa of Mercia (a kingdom situated in large portions of present-day England) had gold coins with what was clearly intended to be Arabic script on them. And this was in the late 700s, so keep that in mind.