I’m so grateful that you’re sharing your faith so deliberately. I know it can be daunting, and sometimes frustrating, especially with someone like to put you on the defensive. I take it from the way you’ve phrased your question that this is what’s happening. He seems to have stacked the deck in his favor and shielded himself from having to prove anything, even to himself.
If he’s demanding that you prove (1) the Quran is false, (2) that Jesus is God, and (3) the Bible is true, is he willing to shoulder the burden himself? Could you demand that he prove (1) The Quran is true (and the word of God), (2) that Jesus never claimed to be God, and (3) the Bible is false? If he’s not willing, then you know that he’s also unwilling to have a fair and sincere discussion.
Consider this approach: First, agree on the ground rules for dialogue. Would your Muslim friend agree that the person who is making a claim bears the burden of proving that claim to be true? That seems to be intuitively sensible in any kind of discussion like this. In other words, it is not your burden to prove that the Quran is false, it is his burden to prove the Quran is true. Actually, as a Muslim, he hasn’t even higher burden. He has to prove that the Quran is the inerrant and perfect word of God and that what we have today has no changes in it since Muhammad spoke it. You see, the entire Islamic faith depends on the divine inspiration of the Quran. It’s one thing for a document to be “true”. That doesn’t mean it came from God. It’s another thing to prove that it has divine origin. There is a lot of evidence that the Quran has been redacted, edited, and changed over the centuries. Plus, there is plenty of evidence that it doesn’t contain scientifically accurate information, quite the opposite actually.
One example of a troubling aspect of the Quran is its denial of Jesus’ crucifixion in Sura 4:157-158. This is a problem because, by far, the great weight of historical evidence demonstrates that Jesus did die by crucifixion. This comes from secular scholars who have no reason to be biased for or against this claim. Your friend bears the tremendous (and insurmountable burden) of proving that Jesus wasn’t crucified and that he didn’t die on the cross
I’ve also found that sometimes (but not always), confident Muslims really haven’t put their beliefs to criticism. They often swallow claims that the Qur’an hasn’t been changed, that they Bible has, that Jesus never claimed to be God, etc. without actually seeing if those things are true. Perhaps you can ask your Muslim friend what he’s done to see if his beliefs are true. What material has he looked at from those who don’t agree with them? What has he found to be the toughest challenges? He may have done this, but he may not have. If not, then it seems to me that his confidence is a mask for insecurity. There are plenty of sites that contain well researched and well reasoned articles calling into question the veracity of Islamic beliefs. If you look at them (for example, www.answering-islam.org) and provide him with the articles, I wonder if you’d start to have a more fruitful conversation.
Underlying all of this is the fact that many Middle Easterners and Easterners, especially Muslims, are immersed in an honor-shame culture in which questioning or doubting one’s heritage brings shame. People get ostracized and criticized just for raising questions. You might want to explore with him what the consequences would be if, for the sake of argument, Christianity were proven true and Islam false. Once he admits that the consequences could be devastating, I’d ask him - aren’t those consequences good reasons to not be open minded about whether Christianity is true? The fact that he seems to be putting all of the burden on you suggests that he may be worried that when he takes the burden on himself to prove his beliefs true, he won’t be able to. Saddling you with that burden relieves him of that anxiety.
In sum, make sure that he’s not putting too much burden of proof on you and is unwilling to take any onto himself. You take on the burden when appropriate. Challenge him on the motivations for discovering if something is true or false. I hope this helps!