This is a controversial question, but here’s how I see it: Muslims believe that their god is Allah, and that Jesus was just a prophet. They claim to worship the god of Abraham, as do Christians. However, Christians believe in the Trinity (a Triune God), where God the Father, God the Son, and God the Holy Spirit are all deity. The question basically comes down to whether one can reject Jesus Christ as the Son and truly know God the Father. And it is Christ himself who answered that question in John 14:6 when He said that to reject the Son means that one does not know the Father: “Jesus said to him, ‘I am the way, and the truth, and the life; no one comes to the Father but through Me.”
So, based upon this, I would say no, that Muslims do not worship the same God as we Christians.