If a Q or the entire Q Continuum, decided to destroy the Universe, would they be able to?
The universe is - per defintion - all of spacetime and everything within that. However in Star Trek, there seem to be things beyond spacetime itself. So the question is if the definition of universe had to be expanded is left open as far as I know.
Qs per se can exisit out of time. This is seen in "All good things..." (TNG S07E25/26). Here, Q more or less tricked Picard in several points of time to create anti-time which - if letting it come to conclusion - would annihilate with time an as thus spacetime itself. (Nonsense from our physics point of view, but in ST-verse the accepted theory and confirmed by Q as a paradox that can exist.)
So even Federation technology is capable of annihilating the universe and Qs are clearly beyond that.
However Qs do have limits as that Qs have been imprisoned by other Qs like Quinn. They can be made into any species and stripped of their powers by other Qs. They can be killed by other Qs like Amanda Rogers parents and they can be harmed by humans that have been enabled to enter the Q continuum and equipped with Q "weapons" as seen during the Q civil war.
Just how far their powers actually go is never exactly explained, but unmaking the universe as in unmaking spacetime is possible. If they can unmake what they are themselves however, which a new definition of "universe" should also entail, is never stated.
The word universe literally means absolutely everything. This would include the Q continuum. In the episode the The Q and the Grey they show the Q continiuum is embroiled in a civil war. If any given Q could wipe out the entire universe they could logically take out all of the Q they consider to be an enemy. The civil war proves they can't do that.
If unopposed a Q might be able to wipe out all of what would be considered our part of the universe - all of the planets, stars galaxies etc. as they claim to be omnipotent and omniscient. However it is unlikely they would be unopposed as there are other omnipotent beings in the Star Trek universe such as the Douwd and Trelane who would presumably block the Universe being wiped out.
The limits to their powers would be whatever someone else, possibly another Q, doesn't prevent them from doing.
Yes, they could.
As shown in various episodes, the Q have the power to create new dimensions, alter the very fabric of reality in any way they see fit and they can even hurt and kill themselves and each other and bring untold destruction on their own continuum.
In Qpid (TNG S04E20), Q creates a temporary pocket dimension based on the story on Robin Hood, with the officers of the Enterprise playing Robin Hood and his Merry Men, much to the dismay of Worf. Q is able to create this dimension, give it an end time and even prevent himself from being able to interfere with it or know what's going to happen.
PICARD: We will no longer share in this pointless fantasy of yours.
Q: Fine. stay here and do nothing. By midday tomorrow, your crew will be safely aboard their ship. Of course, you will have to accept the consequences of your inaction.
PICARD: Q, I ask you to put and end to this before someone gets hurt.
Q: Oh, that's impossible. You see, I've given this fantasy as you call it, a life of its own. I have no more idea what's going to happen than you do. But of one thing I am absolutely sure. If you dare come to Nottingham Castle, blood will be spilt.
In Deja Q (TNG S03E13), we learn that the Q Continuum decided to strip Q of his powers, due to his repeated shenanigans and his ignorance of Q laws, showing that several Q are able to overcome the power of one individual Q.
Q: I haven't the vaguest idea what you're talking about and I have a much more serious problem. I'm no longer a member of the Continuum. My superiors have decided to punish me.
PICARD: And punish us as well, it would seem.
Q: They said I've spread chaos through the universe, and they've stripped me of all my powers. You don't believe me, do you? Do you think I would humiliate myself like this?
RIKER: If it served your purpose, yes.
Q: It's the truth. I stand before you defrocked. Condemned to be a member of this lowest of species. A normal, imperfect, lumpen human being.
TROI: They made you human as part of your punishment?
Q: No, it was my request. I could have chosen to exist as a Markoffian sea lizard or a Belzoidian flea. Anything I wished as long as it was mortal. And since I only had a fraction of a second to mull and I chose this and asked them to bring me here.
The dialogue shows that there is a hierarchy of powers in the Q Continuum, confirmed later in Q2 (VOY S07E19) when Q's son passes in judgement in front of three Q Judges and is condemned to remain human for his actions. Q appeals of the verdict and later explains to Janeway that he had to come to an agreement with the judges.
JANEWAY: So, the Continuum crumbled like a Gelbian sand sculpture?
Q: Yes. Some of them even got on their knees and begged me to stay. It was pathetic. If you must know, I had to agree to a few minor conditions.
JANEWAY: Oh? Such as?
Q: Eternal custody of the boy. Wherever the little brat goes, I go with him.
JANEWAY: I hope he won't spoil too much of your fun.
Still in Deja Q, Q suggests that the crew alter the gravitational constant of the universe in order to move a moon in a safe orbit. Something he could apprently easily do if he still had his powers.
DATA: Can you recommend a way to counter the effect?
Q: Simple. Change the gravitational constant of the universe.
Q: Change the gravitational constant of the universe, thereby altering the mass of the asteroid.
LAFORGE: Redefine gravity? How am I supposed to do that?
Q: You just do it. Where's that Doctor, anyway?
There is also the case of Amanda Roger's parents, killed by the Q Continuum for having a child despite being ordered not to procreate when they decided to abandon the Continuum to live as humans on Earth (TNG S06E06 True Q), Quinn the suicidal Q who was prevented from killing himself by being imprisoned in a comet for the rest of eternity, until Voyager inadvertently broke him free (VOY S02E18 Death Wish), and the Q Civil War (VOY S03E11 The Q and the Grey) where we can see many Q, including the one we know best, be injured by their own weapons in the illusion based on the US Civil War.
Thus, the Q could very easily destroy the universe, every dimension there is, and themselves, if they so wished.
A lot of decent answers are here, but I will add one possible interpretation that was not covered:
We obviously cannot know for sure if they could destroy the universe, there is not enough information to make a solid judgement. We can however speculate on the matter.
As far as individual Q, such as the Q (John De Lance) that we know from TNG, probably not. He seems to have a sort of "nigh-omnipotence" and/or at the very least high tier reality warping. That said, those powers appear to be overshadowed by the Continuum itself, and possibly some other individual Q's powers.
Now however looking at the Continuum as a whole, I would argue probably yes. If you think of them in the way of the Borg, in that working and utilizing their powers together, you get something much greater than any individual could accomplish. Thus, this would magnify their already massive power to an amount unfathomably higher than that. This would allow for some very "Omnipotent-Like" powersets, and likely the ability to do whatever they wanted with the Universe.
Basically that's it, given what we have seen so far in Star Trek canon, I believe this to be true. Mind you, it is always possible that there exists something even greater than the Q, and while they might be able to destroy the universe, this greater faction/entity might not allow it. I find that unlikely however as the Q Continuum have been fairly consistently shown as the top power in the Star Trek universe.