No, the God of Christianity and the God of Islam are not the same. First of all, the God of Christianity is a Trinity where the God of Islam is not. The Trinity is the monotheistic teaching that God exists eternally as three distinct persons: the Father, the Son, and the Holy Spirit. In Islam, this is blatantly denied.
"And behold! Allah will say: "O 'Isa Ibn Maryam! Did you say to men, "Worship me and my mother as gods besides Allah?" He will say: "Glory to you! Never could I say what I had no right to say. Had I said such a thing, you would indeed have known it. You know what is in my heart, though I know not what is in yours. For you know in full all that is hidden," (Quran 5:116).
"O People of the Scripture, do not commit excess in your religion or say about Allah except the truth. The Messiah, Jesus, the son of Mary, was but a messenger of Allah and His word which He directed to Mary and a soul [created at a command] from Him. So believe in Allah and His messengers. And do not say, "Three"; desist - it is better for you. Indeed, Allah is but one God. Exalted is He above having a son. To Him belongs whatever is in the heavens and whatever is on the earth. And sufficient is Allah as Disposer of affairs," (Quran 4:171).
Furthermore, in Christianity the doctrine of the Trinity allows for the incarnation of the Word. John 1:1, 14 says that the Word which was God was with God and became flesh and dwelt among us and was crucified (Matt. 26:2; 27:38). This is denied in Islam, which says that Jesus is only a prophet and was not crucified.
"[Jesus] said, "Indeed, I am the servant of Allah . He has given me the Scripture and made me a prophet," (Quran 19:30).
"And [for] their saying, "Indeed, we have killed the Messiah, Jesus, the son of Mary, the messenger of Allah ." And they did not kill him, nor did they crucify him; but [another] was made to resemble him to them. And indeed, those who differ over it are in doubt about it. They have no knowledge of it except the following of assumption. And they did not kill him, for certain," (Quran 4:157).
So, the God of Islam and the God of Christianity are not the same.