I have learned that the roles and expectations about religious beliefs vary quite a lot among and inside families. That is one reason why it can be such a difficult subject.
How best to deal with your uncle will, in my opinion, depend mostly on how much space he has with you. It helps that your mother is not encouraging him.
To the extent that it is confortable, it may help to avoid the subject matter. But I am not generally speaking very sympathetic to strategies of deception, even by avoidance of a subject. Having been raised by a couple of fanatics in Kardecist Spiritism, I have learned all too well that confrontation may be necessary, if only to avoid being an unwilling accomplice or victim.
Maybe you could ask your mother whether it could be a good idea to state to your uncle once, politely but clearly, that you are not willing to listen to any proselitism for Islam? That may be a delicate thing to handle, since it is after all her brother. But there is a very high chance that she will have no choice either.
Since he is insisting in specifically asking you to read the Qur'an, you may want to do just that to your own satisfaction (not necessarily much of it), so that you can challenge his claims if it comes to that. Having actual facts from which to argue is usually very helpful, and there seems to be a considerable undercurrent of hope that the Qur'an will somehow magically turn people into Muslims if they only are decent people who begin to read it with honest intent. Such hopes are IMO very misguided, but each and every time that you say or suggest that you never laid eyes on its contents you are passively encouraging their survival - and I doubt that to be a good thing for any of you.
Ultimately, I advise you to consider the simple statement that "Uncle, it is not within your rights to demand me to share of your believe in the Quran. Even the Quran itself says that there is no compulsion in religion, so please stop giving me compulsion." See if you find it both truthful and confortable. Tinker with it as you see fit, until you feel confortable to state it clearly and sincerely to your uncle. It may well be necessary, although I must make it clear that there is no telling whether the results will be pleasant.