No. In the movie novelization, it was revealed that this being had been imprisoned there, leaving the crew wondering, "what being was so powerful that it could have imprisoned this one!"
That answer was never revealed. All that was revealed was that the being wasn't "God", just a being that was able to touch Sybok's mind. It claimed to be God, because that's what would motivate Sybok to seek it out, giving it a chance to escape.
However, in non-canon books...
According to the non-canon Q Continuum series of novels by Greg Cox,
the Sha Ka Ree entity was known as "The One," and was exiled to the
center of the galaxy for causing a civil war centered around the
premature destruction of the legendary Tkon Empire millennia before
being rediscovered by Sybok and the crew of the Enterprise-A. He was
an associate and contemporary of 0 (pronounced "Nil", as in nothing),
the Beta XII-A entity, or (*) as 0 referred to it (used in type
because its real name was apparently unpronounceable, or even
incomprehensible, by Humans), and Gorgan in this endeavor.