How do you witness to a Muslim?
The same way you witness to anyone. Like the Apostle Paul’s great example, you start where THEY are at.
To the Corinthians, Paul didn’t start spouting a bunch of Jewish history, prophecies and facts that had no clue about. He start with the SIN they experienced in this culture and the “gods” they worshipped: gross immorality, paganism, etc … and convicted them of their sin.
To the Jews, he started with Scripture and expounded because they already had a Biblical worldview.
Forty years ago in America, we would start by simply talking about “Christian” things: salvation, church, Jesus, etc. That is no longer effective because that foundation doesn’t exist anymore in our country. So now we have to approach Americans like Corinthians… you start with where our society is today: immoral and ungrounded in Biblical understanding.
With the Muslim, you start where he is at too. You can’t start spouting “Christian’eze” because they have already been taught that Christians are wrong, corrupt and infidel unbelievers who worship a pantheistic god.
In all cases though, the path is the same: convict a man of his sin, and then give him the answer. The answer (salvation) is unnecessary until they believe they need it (eternal condemnation because of their own sin). Remember this phrase at all times: THE LAW CONVICTS, THE GOSPEL SAVES.
No matter their background, a person doesn’t want to hear the Gospel until they understand they have a need for it. Otherwise it is foolishness, a stumbling block. The Law, the 10 Commandments, teaches a person they have sinned and are justly condemned. Until they reach that point, the Gospel is useless.
To pursue arguments about doctrine, the Koran, etc may open some doors in conversation, but the real effort for any and every opportunity to evangelize is simple: conviction of sin, presentation of the solution. The Law convicts, the Gospel saves.