The Jewish God is the Christian God.
If we say just because Jesus Christ is not mentioned, Holy Spirit not working in the lives of the people in the Old Testament times and for that reason the God of the Old Testament is not the God Christians worshipped. then its terribly wrong doctrinally.
In the Old Testament, we will read about the Spirit of God moving over the waters(Genesis 1) and in the lives of the prophets and kings God ordained, He send the Holy Spirit to rest upon them to empower them, though for common people this is not the case. We also read about Jewish rites of sacrificing animals and other reference scriptures that foretold the coming of the Messiah. The God of Abraham, Isaac and Jacob is the God we worshipped because all the promise of blessings given to Abraham by God Himself will soon be fufilled in their descendant, Jesus Christ - Son of David.
Therefore, we see a flow from the Old to the New Testament. The God who blessed Abraham and promised him that out of his descendant will come a blessings for all nations. Through the line of Isaac(not Ishmael) will one offspring comes which will be a blessing for all people. Jesus Christ coming to this earth brings this covenant to a fufilment as well as many other prophetic scriptures. The God of Abraham, Isaac and Jacob has now come in human flesh in the person of the man Jesus Christ. He is fully God and fully Man. The wholeness of God dwells in Him and He is full of grace and truth. The Word which from the beginning with God and is God now made flesh and dwell among us. Jesus still lived the typical Jewish man's life. He observed the Feasts of the LORD and other jewish customs. He teach from the law of moses, He speak about the Old Testament and He Himself being the fufilment of the promise. He is the Messiah the Jewish people have been looking forward.
With that, the Jweish God though still worshipped by the Jews in the Synagouges, is the God we read about in our Bible(Old and New Testaments).
The doctrines of trinity though not expressively taught in Judaism does not mean that the trinity does not exists. It is only in Jesus Christ can the fullness of God be revealed to us. There is no contradicting in any doctrines except that people of Judaism refuse to believe Jesus being the Messiah since He came and die for them on the cross and appear like a looser. The Jews, in their pride and expectations expect to see a powerful warrior as their Messiah, someone of social and political status.
However, more and more Jews have come to realised that Christ is indeed the Savior of the world. We call them Messianic Christians.