Q was pretty clearly designed as an antagonist and foil for Jean-Luc Picard, and he never made a lot of sense in the other series. He was brought into those series, like the Borg, for ratings stunts. And it showed.
Thankfully, the writers of Enterprise never panicked and pushed the Q button the way Voyager's did. The Borg make a questionable and continuity-challenging [1] appearance on Enterprise, but Q never does.
I watched all of Enterprise. It...gets better in Seasons 3 and 4. That's about all I can say. The first season is pretty bad. Season 2 has a few good episodes. In Season 3, things start to pick up a bit, and Season 4 is legitimately interesting. The show never quite reaches the heights of TNG, DS9, or TOS. But it gets pretty good. I'd only recommend watching it all the way through if you're ready to wade past the early crap, though.
[1] This is kind of a mixed bag. The explanation for their appearance stems from the events of Star Trek: First Contact, in which a small detachment of Borg drones are left behind on Earth in the 20th Century. So the timeline has been altered a bit, and hence, those remnant Borg show up in Enterprise. On the flip side, you'd think that an early encounter with the Borg would have stuck in everyone's memory a little bit, and that Picard wouldn't have been shocked upon first encountering them. The episode acknowledges the issue, at least, and attempts to arrive at a workaround. The fix is clever, if not entirely satisfactory.
Ugh, don't remind me. That was the worst. Why can't shows just acknowledge that makeup and effects change over time, and leave well enough alone? Augment Virus was Enterprise's attempt to cave into fans' demands that Star Trek offer an explanation for the TOS Klingons. If you ask me, Star Trek should've just hand-woven that subject indefinitely.
DS9 handled the matter much more effectively in "Trials and Tribbleations." Someone asked why the old-school Klingons had smooth foreheads, and Worf replied "We don't like to discuss that with outsiders." Boom. Perfect explanation. Funny, clever, and hand-wavy. Done.
Unfortunately, that joke on DS9 (which I absolutely love) was taken as establishing as canon that TOS-era Klingons actually looked different. A sane person would interpret it as a meta joke in an episode full of them, but these are Star Trek fans we're talking about here.
That still doesn't justify Enterprise trying to explain it. If one must insist that there be an in-universe explanation because of the DS9 precedent, just say Q decided to mutate the entire species for a hundred years as a lark.
The classic "a wizard did it" defense. :) Come to think of it, that's just about the only potential appearance of Q in Enterprise I would have tolerated. "Oh, they look different? Hmmm. Q did it."
... right ? i don't know, i watched only a few episodes of enterprise