That's an intriguing question. I'm interested in planetary formation from the standpoint of an armchair amateur, but the proper answer is probably above that paygrade.
The gas planets are thought to form at a stage in the process where the stellar nebula of gas and dust still has a substantial mass. The dust accumulates into clumps, through collision then gravitational attraction, which build up to form planetesimals, which accrete to larger bodies which may go on to become asteroids, moons, or planets. But some become the core of the gas and ice giants that attract the remaining gas within the nebula. These planetary cores are thought to be five or ten time the mass of the Earth.
I find it unlikely that these could then wind up as a moon - they would just be to big - but that may simply be lack of imagination on my part. And I don't think they could accumulate significant gas without being massive to begin that process. On balance, then, highly unlikely, but in a rather large universe I hesitat to say impossible. Fortunately there are others on the forum who likely know the answer.
