Dear Hadi,
Dear Wondering about the Jewish God,
We're not sure where you came to your understanding that Muslims worship a Jewish God, but let us try to help clarify.
As Muslims, we believe that there is one Almighty God and that He created the Universe and all that is within it, including all of humanity, to which He revealed one message.
We believe that this Message was revealed through a long line of Prophets and that the message took different forms as humanity progressed through different stages in its history.
The essence of this message, which unites all monotheistic faiths, is to worship God and only God and to live in this world, submitting to this reality that He is the Creator and having this Message guide our conduct.
This unity of the Message is clearly stated in the Quran 2:136,
Say: "We believe in Allah (God), and the revelation given to us, and to Abraham, Isma'il, Isaac, Jacob, and the Tribes, and that given to Moses and Jesus, and that given to (all) prophets from their Lord: We make no difference between one and another of them: And we bow to Allah (in Islam)."
Therefore, Muslims do not worship a "Jewish God," but rather, we believe that Muslims, Christians, and Jews all worship the same God.
In peace.