As shown in various episodes ("Mirror, Mirror," "Phantasms," etc.), other universes exist, oftentimes containing the same characters, albeit in different situations than in the prime universe. I'm curious, though, whether individuals that apparently exist outside of the multiverse, such as members of the Q, would not be duplicated elsewhere. It seems reasonable to me, but I'm not sure.
Q has demonstrated the ability to transcend multiple time-streams in the episode TNG: All Good Things where
Picard inadvertently destroys the human race through the creation of a temporal paradox.
It also appears that the Q continuum itself exists outside the normal "multiverse" seen within the Star Trek universe with Quinn noting that the "dusty road" (seen in VOY: Death Wish) leads to the universe.
He's definitely the Q we know from The Next Generation, correct?
MJ: Indeed. The Q Continuum exists outside of the alternate timelines that our tiny flesh-brains occupy. Every Q is the same Q across all parallel realities, and this Q maintains both his fascination with and his contempt for the human species in particular.
Is this technically set before Q has met Picard and the others in TNG?
MJ: After. In this new story, Q has already had all of the adventures we've seen in the television series over the years, and he's aware that Spock Prime (Leonard Nimoy) traveled from the old timeline to the new one.
So does this mean there['s] a Q in the new-Trek universe, or is there only the one?
In-Universe Reference texts
The Star Trek Encyclopedia directly refers to the Q as being extradimensional rather than existing within any specific dimension or universe:
Q, (John DeLancie) : An immensely powerful extradimensional entity. While possessing near-godlike powers. Q also exhibits a childlike petulance and sense of playfulness
I think that Q goes beyond just the normal universes. In Star Trek Voyager he goes back to just before the Big Bang of our own universe. So I don't think that Q would be duplicated elsewhere.