I, as well as Lula, do not believe the Islamic God (Allah) is the same God of the bible. I do not believe the Jews' God and the Arabs' God are one and the same.
Hate to disappoint you. It's the same god.
Abraham taught both his children about the one true god, and both children and their descendants believed. Muhammed was a descendant of Ishmael and believed in the same god as Isaac and Ishmael did. So far none of this has anything to do with "Islam" in the Muhammedan sense. Ishmaelis believed in the same god as the Israelites.
There were many gods Semitic tribes believed in, among them the Creator, the only real god, the god ultimately known by His name and by "El" (Alpha Lamed, also simply means "god") and "Elohim" (Alpha Lamed Heh Yud Mem, this is a plural of Aleph Lamed Heh) in Hebrew and "Allah" (short for Aleph Lamed Aleph Lamed Heh, with the first Aleph Lamed being the definite article and Aleph Lamed Heh being the word) in Arabic. Hebrew-speaking Jews refer to Him as "Elohim", Arabic-speaking Jews, Muslims, and Christians refer to Him as "Allah".
At some point Muhammed, the Ishmaeli, believed he was a new prophet. His prophecy, later written down in the Qur'an (from the root Quf Resh Aleph, meaning "read" or "happen" with "Qur'an" meaning "recitation"), confirmed many of the stories in the Bible, not because G-d told Muhammed, but simply because Muhammed already believed in Adam and Eve, in Noah, and in Moses and the exodus.
However, the god Muhammed believed in and the god Muhammed thought he was a prophet of is that same god Jews and Ishmaelis (and many other Semitic tribes and peoples) and Christians worshipped.
I don't know where the legend that it's not the same god comes from, but it's simply not possible because a) Abraham taught both his children about the one true G-d,
the words in Hebrew and Arabic simply don't allow for another interpretation, and c) there is no other (real) god, which means that even if Muhammed was not a prophet, the god he THOUGHT he received his prophecy from MUST have been the same god as the Jewish and Christian god.
However, there is another Semitic people (or sect) called the Mandaeans who also believe in Adam and Eve and Noah but not Moses and the exodus (don't know their stance on Abraham) and they believe that the god of Adam and Noah is NOT the god of Moses, Jesus, and Muhammed (but I think they do believe in John the Baptist as a prophet but I might be wrong on that). The Mandaeans speak Aramaic, I think. I believe "Manda" means "knowledge" in Aramaic. (In Hebrew "Mada" (Mem Dalet Ayin) means "knowledge", the missing "n" could be a regular consonant shift, I don't know.)
So if you ask me if I believe in the one true god, the Semitic creator god "El" or "Elohim", the god of Abraham and the god of Moses, I will say that, yes, I do.
But if you believe in that same god or not, I cannot tell. Abraham did, Isaac did, Ishmael did, Moses did, Muhammed did, and I do.
I really don't care if you worship the same god or not (who cares what gods other people worship anyway, as long as such worship doesn't involve ugly sacrifices?), but I won't let you modify my G-d just because you want to exclude one half of Abraham's children and show such disrespect to Abraham.
If you worship a god who is not spelt Aleph Lamed Heh and who is not the god of Ishmael, you are not worshipping the same god as Isaac and Moses.
You can clearly read in the Bible that Abraham's covenant with G-d covered both Isaac and Ishmael (who was 13 at the time). In fact, I seem to recall that Isaac was even born when the covenant was made.
I assume you both know what the sign of that covenant was and is and that it is practiced by Jews as well as Ishmaelis and all who follow the religion founded (or re-affirmed, depending on what you believe) by the most well-known Ishmaeli after Ishmael.
These things are not coincidence.
You must stop using the word "Allah" as if it were a name. "Allah" is simply the Arab word for "god" (with a definite article attached, as is proper per Arab grammar). Arabs and Jews happen to refer to god by the same word because Arabic and Hebrew are so closely related. If "Allah" is different from "god", then so is the Hebrew "Elohim" because it's the same as the Arabic.