Is the God of Muslims the same as the Christian God?

The answer is no. In the case of Catholics, Protestant’s, Orthodox and many others the answer is no In the denominations where the Trinity is worshiped the Father is God, the Son is God and the Holey Spirit is God in a Trinity. In Islam there is no trinity. God is ONE. Muslims worship the One and Only God, the creator of all things as the Jews do and all the prophets including Jesus (pbuh). 

Some Christians denominations do not worship a Trinity, they worship God the creator but they do believe that God as a Son. In Islam, God has no offspring and no partners.