The God that communicated to Moses and the prophets of the Old Testament is the same God who was finally and fully revealed in the person of God's Son, Jesus Christ.
Hebrews 1:1-2 wrote:
1 God, after He spoke long ago to the fathers in the prophets in many portions and in many ways, 2 in these last days has spoken to us in His Son, whom He appointed heir of all things, through whom also He made the world.
Anyone (Jew, Moslem, Hindu, or even "Christian") who rejects Jesus is rejecting God and, whatever that person may ostensibly be doing, he is not worshipping God. (I include "Christians" in the list because many who identify themselves as Christians are not truly born again and some "Christians" do not see that Christ is essential to having a relationship with God.)
John 5:19-23 wrote:
19 Therefore Jesus answered and was saying to them, “Truly, truly, I say to you, the Son can do nothing of Himself, unless it is something He sees the Father doing; for whatever the Father does, these things the Son also does in like manner. 20 For the Father loves the Son, and shows Him all things that He Himself is doing; and the Father will show Him greater works than these, so that you will marvel. 21 For just as the Father raises the dead and gives them life, even so the Son also gives life to whom He wishes. 22 For not even the Father judges anyone, but He has given all judgment to the Son, 23 so that all will honor the Son even as they honor the Father. He who does not honor the Son does not honor the Father who sent Him.
That Moslems worship Allah who is their one one and only God does not mean that they are worshipping the One God who as existed from eternity as the Triune God, Father, Son and Holy Spirit.