In a sense, that was touched on in TNG's "Emergence", when the Enterprise got self-aware due to some storm and it created a new lifeform.
I have wondered what happened to it, though...
Frell, the original Enterprise became self-aware in "The Impractical Joker."
The crew ended up getting tired of it and ran it through the anomaly that made originally the computer awaken, to undo the process and return the computer to normal.
Trek has a very conflicted relationship with AIs in general, so it's nothing personal...