If I recall correctly, it really wasn't explained how or why Q was dying in Star Trek: Picard. All that was said about it was that he had believed his species was immortal, and that his deterioration was something he had previously thought was impossible.So Q could still be alive, or the Continuum could bring him back.
Season 3 Picard spoilers
But after the point where he apparently went away to let his death come and go, he showed up again at the end of the following season somehow alive and well with no solid explanation. When questioned about how he wasn't dead at a point when he was supposed to be, he merely answered, "Oh, and here I was hoping the next generation wouldn't think so linearly." So this later appearance from Q was either a version of him from an earlier point in time before he died, or his death doesn't work in the same linear and finite fashion that ordinary mortal deaths do. The writers intentionally left the true nature of how death works for him and his species completely vague and mysterious.
As for the other Q from the Voyager episode who wanted to die (and renamed himself "Quinn"), he was permitted to be separated from the Continuum and made into a mortal humanoid being before he died, so his death was just an ordinary one with nothing vague or mysterious about it (as he was no longer a Q when it happened).
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