I would argue that Muslims don't even worship a God at all and that Islam is not even a religion, but a political ideology.
Mohammad claimed to have revelations, but there are also verses in the Qu'ran that basically say things like, "Allah says when you visit the prophet (Mohammad) for dinner, leave promptly when it's done because it annoys the prophet when you stick around after dinner (because he wants to have sex with all nine of his wives tonight)." Obviously that's not an exact quote, but there are other examples of things like that which support the conclusion the Mohammad wrote the whole Qu'ran speaking as himself but pretending to be a god.
For more information on this specific example, I recommend David Wood's explanation: https://www.youtube.com/watch?v=HgKn_OaDOHI&list=WL&index=9&t=0s
I also recommend David Wood's YouTube channel, Acts17Apologetics for more information about Islam from a Christian perspective.
I would also mention that the Qu'ran tells Muslims to kill Christians and Jews, and it permits polygamy, FGM, and various other things. As a Christian, the God I believe in does not permit or encourage murder like the "god" of the Muslims does, so I would say no, we don't believe in the same God. I know that most Muslims aren't evil like their founder was, and I get that some of them are actually very reverent and respectful to who they believe is God, but going back to the beginning, Muhammad was an evil, murderous, polygamist, pedophile child rapist who founded a "religion" in order to make money (the Qu'ran orders taxes to be paid by subdued Christians and Muslims) and give men unlimited sex (it approves of polygamy and promises virgins to men who kill people for Mohammad).