There are hard ways to answer this question and there is an easy way. Since Jesus is the anchor of the easy way, of my faith and describes Himself as the Way, I am taking the easy way.
No, by definition, Christians and Muslims do not worship the same God. Jesus makes that claim impossible.
According to the Bible, Jesus is God. Jesus is eternally existing God and through Him and by Him all things were made. Jesus is a person of the Trinity, three persons, one God and it is impossible for Jesus to have been created, for Jesus to be separated from the Godhead and for God to be defined without the inclusion of Jesus and the Holy Spirit. Without Jesus, there is no God according to the Bible.
Islam denies that Jesus is God, that Jesus is eternally existing God and that through Jesus and by Jesus all things were made. Islam defines their god as not being a triune god and says that Jesus was just a prophet. Prophets are great and all but they are not God. Without Jesus, Islam claims that you are fine, Allah is god separate from Jesus.
Are we getting the picture? The two sets of beliefs are completely inconsistent because of Jesus and cannot define the same God. It is like truthfully saying that your friend Bob is 7 feet tall and has red hair and another person saying that they know your friend Bob and he is 5 feet tall and has blue hair, not the same Bob.
For Christians, Jesus is God or God is not God. For Muslims, Jesus is not God or God is not Allah. It is not the same Bob. Simple theology, perhaps, but it is also clearly true.
Just a thought for the day.