Is the God of Islam, Christianity, and Judaism the same God?
The short answer to this question is, Absolutely not! If you want to dig into why I encourage you to read to the end.
The Source
The question boils down to one fact, our source of information about God must come from His self-revelation given to us in His self-attesting Bible. This revelation must exclude all other sources of supposed revelation and information and must include the entire Bible and not just portions of it. Otherwise, we are starting with human opinion and human reasoning. Both are filled with error and are fallible.
A human being cannot self-attest to know ultimate truth. The Bible, however, is the only reliable self-authenticating source of Truth. We must start there to answer whether Islam, Christianity, and Judaism worship the same God.
The god of Islam
Islam claims to accept the Torah, the Prophets, and the Gospels. However, according to Islam, the Bible must be interpreted through the lens of the Quran. The problem with this is that Islam rejects that God is the self-existing Trinitarian God of the Bible. Islam rejects the full deity of Jesus Christ, the Second Person of the Trinity. It denies the teaching given throughout the Bible that salvation is by faith alone, in Christ alone, through His blood alone.
These are not the only essential doctrines concerning the one and only True God rejected by Islam, but they are sufficient to show that the god of Islam is not the same as the self-existing Trinitarian God of the Bible. Add to this that the Quran claims authority over the Bible, and it is easy to see that the god of Islam is a false god.
The one and only Trinitarian God is clear when He writes in Exodus 20:3:
Thou shalt have no other gods before me.
Adding to and Taking Away
The Holy Spirit condemns both adding to and detracting from the "prophecy" of Scripture in Revelation 22:18+19:
I warn everyone who hears the words of the prophecy of this book: if anyone adds to them, God will add to him the plagues described in this book, and if anyone takes away from the words of the book of this prophecy, God will take away his share in the tree of life and in the holy city, which are described in this book. (emphasis mine)
Islam both adds and takes away from God's self-attesting revelation given in the Bible.
The god of Judaism
Judaism does the same. While Judaism does not recognize the Quran as divinely inspired, it includes the Oral Torah in its canonical books. This consists of the Mishnah, the Talmud, the Halakhic Literature, Responsa, and the Codes. Judaism, therefore, adds to the Bible just as the Quran.
Judaism also "takes away" the self-revelation of the one and only self-existing Trinitarian God by rejecting the entire New Testament. While God gave us glimpses of both His Unity and the Trinity throughout the Old Testament, we see the fullness of the Trinity revealed throughout the New Testament. What were shadows and types in the Old Testament God reveals fully in the New.
Other gods
The "god" of Judaism is not the self-existing Trinitarian God; the one God existing in three Persons, Father, Son, and Holy Spirit. Any god that doesn’t begin and end with God's revelation given in the Bible, is a false god. This includes the gods of Mormonism, Jehovah's Witnesses, Liberal and Neo-Orthodox Christianity, Hinduism, etc.
I include Liberal and Neo-orthodox Christianity, along with all paracytic claims to Christianity, because they do not hold to the inerrant and infallible revelation of the self-existing Trinitarian God given in His self-authenticating Bible. If my reader wants to dig into this deeper, I encourage you to start with an excellent book written by J. Gresham Machen called Christianity and Liberalism. Machen’s critique of Liberal Christianity can be applied to the various “theologies” masquerading as Christianity in our day including but not limited to Liberation Theology, Feminist Theology, Queer Theology, Word of Faith, Purpose Driven, Seeker Friendly, etc.
Conclusion
In summary, the one and only True God has revealed all that we need for faith and practice in His self-authenticating Word. Any religion that rejects the Bible as their only source of revelation about God or adds other revelation to the Bible are false religions. Our information about God must come from God Himself and nowhere else. Any other sources are false sources, and any other gods are false gods.