As stated earlier, Trelane was a child with a machine that were his "training wheels". His parents were very apologetic to Kirk for his mischief, which seems to be beneath the Q to ever do. So I don't believe them to be of the continuum.
The aliens of pure energy that gave Charlie his power to survive, seem very similar to the Organians, also of pure energy, who are compassionate, but prefer to be left alone and not interfer with lesser evolved species. They don't seem to fit the continuum mold either.
Apollo was actually Apollo. An alien who, with other aliens and had used machines for their power, were mistaken to be gods in ancient times. They appear to be gone (deceased?) now, so not of the continuum.
The Metrons, who pitted Kirk against the Gorn captain on a planet they created when intruders entered their part of the galaxy, seem to be the closest to the Q in attitude and extent of their power. But were surprised by Kirk's act of mercy. And had said humans were still a savage but promising species, but aslo stated perhaps in a thousand years they could make contact with humans. But they had stated they were the Metrons, not Q, so I don't believe them to be of the continuum either.
The Q was an invention for writers of Star Trek: The Next Generation, and had nothing to do with TOS at all.
But it is fun to compare the omnipotent characters that are out there.