Well, if you take the biblical account literally, Ishmael—the forefathers of all Arabs—moved with his mother, Hagar, southeast into the Arabian peninsula, where over the next two thousand years or so, it evolved into something different, which is known as Arabic.
But that presupposes that Arabia was empty of all human inhabitants at that time, which is highly unlikely, since the Arabian peninsula, along with the Sinai, was the first land that humans spread into from Africa some 250,000 years ago—and again around 70,000 years ago—and has been populated continuously ever since.
In reality, Arabic evolved in parallel with “Hebrew” (which is just the name that we now call the language spoken by all peoples of the Holy Land in the second millennium BCE, but which at the time was simply known as “Canaanite”) from a common Semitic source, somewhere in the Arabia-Mesopotamia-Canaan region.
So if Ishmael moved with his mother to Arabia, he and his descendants would simply have picked up the local lingo within a generation or two.
(Originally written in reply to a question at Quora.com).