Even if Trelane were biologically godlike instead of getting his power from a special mirror (lol), I don't see why he'd necessarily be tied to Q. There's other species like the Daud or even, to an extent, Apollo's race who have similar abilities.
Their ideologies don't even seem to be the same - Trelane is a dick but never hurts anyone (and arguably doesn't intend to, even when hunting Kirk), and his parents are furious at him for messing with other lifeforms. Meanwhile Q throws the Enterprise into Borg space and watches as 18 people get dragged away and assimilated, then makes Picard beg for mercy, which feels a little bit different from Trelane's mischief, to put it mildly.
SNW's decision to make him a Q did inordinately annoy me not because I care that much about Trelane, but rather because it speaks to a certain mindset. TOS' approach is "space is operatic and mythic, anything can happen, there's all kinds of unknowable weirdness out there" while SNW's ethos feels more like "the galaxy consists chiefly of things James Kirk saw, many of which are connected to each other", and the former approach to the setting seems so much more interesting to me.