Why I believe that the God of Quran is the same as the Bible is for many reasons among them those related to Jesus and how the Quran handle it:
1- The God of the Quran declared that he is the one who sent Jesus:
'I have come to you, with a Sign from your Lord, in that I make for you out of clay, as it were, the figure of a bird, and breathe into it, and it becomes a bird by God's leave. And I heal those born blind, and the lepers, and I bring the dead into life, by God's leave; and I declare to you what ye eat, and what ye store in your houses. Surely therein is a Sign for you if ye did believe. (I have come to you) to attest the Torah which was before me. And to make lawful to you part of what was (before) forbidden to you. I have come to you with a Sign from your Lord. So fear God, and obey me. It is God Who is my Lord and your Lord; then worship Him. This is a Way that is straight.'" (Quran 3:48-51).
This is typical if the God of the Quran is the same as the God of Bible
2- God in Quran praised Jesus and his mother Mary. Which is typical if God is the author of the Quran:"Behold!' the Angel said, God has chosen you, and purified you, and chosen you above the women of all nations. Mary, God gives you good news of a word from Him, whose name shall be the Messiah, Jesus son of Mary, honored in this world and in the Hereafter, and one of those brought near to God. He shall speak to the people from his cradle and in maturity, and he shall be of the righteous. She said: "My Lord! How shall I have a son when no man has touched me?' He said: "Even so; God creates what He will. When He decrees a thing, He says to it, 'Be!' and it is." [3:42-47]3- God in the Quran made it cleat the nature of Jesus as a prophet no ambiguity. Which is expected if the true God is the author of the Quran. He did not compromise with that since the start of the Koranic revelation or its end.
"The Messiah (Jesus), son of Mary, was no more than a Messenger before whom many Messengers have passed away; and his mother adhered wholly to truthfulness, and they both ate food (as other mortals do). See how We make Our signs clear to them; and see where they are turning away!" (5:75).
"Such was Jesus, the son of Mary; it is a statement of truth, about which they vainly dispute. It is not befitting to the majesty of Allah, that He should beget a son. Glory be to Him! When He determines a matter, He only says to it, ‘Be’ and it is" (Quran 19:34-35).
4- God did not accuse Jesus for this misunderstanding. He asks him to have his answer clear to Christians in the Quran:
And when God says: O Jesus, son of Mary! Did you say to mankind: Take me and my mother for two gods beside God? he says: Be glorified! It was not mine to utter that to which I had no right. If I used to say it, then You know it. You know what is in my mind, and I know not what is in Your mind. You, only You, are the Knower of Things Hidden. I spoke to them only what You commanded me, (saying): Worship God, my Lord and your Lord. I was a witness of them while I lived among them, and when You took me You were the Watcher over them. You are Witness over all things. If You punish them, they are Your slaves, and if You forgive them (they are Your slaves). You, only You are the Mighty, the Wise. (5:116-118)5- God made it clear about his anger of people taking a prophet as his son and worshipping him.
"Surely, disbelievers are those who said: ‘Allah is the third of the three (in a Trinity).’ But there is no god but One God. And if they cease not from what they say, verily, a painful torment will befall the disbelievers among them (Quran 5:73).
"People of the Book (Jews and Christians)! Do not exceed the limits in your religion, and attribute to God nothing except the truth. The Messiah, Jesus, son of Mary, was only a Messenger of Allah, and His command that He conveyed unto Mary, and a spirit from Him. So believe in Allah and in His Messengers, and do not say: ‘Allah is a Trinity.’ Give up this assertion; it would be better for you. Allah is indeed just one God. Far be it from His glory that He should have a son. To Him belongs all that is in the heavens and in the earth. Allah is sufficient for a guardian" (Quran 4:171).7.6- God had sent correction, this correction is the Quran.22:16 Thus We reveal it as plain revelations, and verily Allah guideth whom He will.56:80 A revelation from the Lord of the Worlds.38:29 (This is) a Scripture that We have revealed unto thee, full of blessing, that they may ponder its revelations, and that men of understanding may reflect.7- Muhammad (PBUH) and Muslims love Jesus and his message.Prophet Muhammad emphasized the importance of Jesus by saying: "Whoever believes there is no god but Allah, alone without partner, that Muhammad is His messenger, that Jesus is a servant and messenger of God, His word breathed into Mary and a spirit emanating from Him, and that Paradise and Hell are true, shall be received by God into Heaven. [Bukhari]That is what you expect if Jesus and Muhammad (peace be upon them) were sent by the same God.