The answer could vary depending on what exactly is being asked.
Do Christians and Muslims (and Jews) describe God the same way? No, though there is some overlap.
Do the three groups worship God in the same way, or teach the same commandments and practices? No, though again there is overlap.
On the other hand…
Are the three religions historically related, with Christianity descending from Judaism and Islam from bits and pieces of both? Yes.
Do the later religions claim to be worshipping the same God as the earlier ones, even if the earlier ones say they are getting it wrong? Yes.
Do they tell some of the same stories about God and His interaction with humans? Yes. Christians say that Jesus is the incarnation of the same God that appeared to Abraham, and Muslims aver that Allah was Abraham’s God and the one who sent Jesus.
So…
We are not required to accept that Muslims are describing God correctly or worshipping Him rightly. We are certainly not required to believe that their religion is truly divinely revealed or salvific. But even purely historically there is enough connection between the two religions and their stories (with Judaism as the father of both) that we cannot deny they are trying to talk about and worship the same God, much as the Jews would have to acknowledge that about us even as they think our notions of the Trinity and the Incarnation are bizarre or even blasphemous. And that gives us a starting point for discussion with the other “Abrahamic” religions that we don’t have with other faiths that lack that connection.