People ask this question in two ways. Some ask as though it is an especially hard question to answer, full of nuance and very difficult to discern. Others ask it as a genuine question, wondering whether Christians and Muslims really do worship the same God.
But really, the question doesn’t require much nuance and has a simple answer. We get some form of this question (or a statement affirming it) frequently in our Muslim-Christian dialogue. Here is how the discussion usually goes:
“I think we worship the same God.”
“Do you think Jesus is God?”
“We do. So we worship different gods.”
There are lots of other things we could say. The questioner sometimes wants to press further. But, ultimately, this cuts through the issues. Christians believe in a triune God. Jesus Christ is the eternal Son of God incarnate. He is fully God. Muslims believe none of that.
So, no. We don’t worship the same God.