This Q, for today's purposes known as Quinn, has been locked up for attempted suicide. By the 2070's, the immortal gadabout had been, seen, and done everything he ever wanted and his society, the Continuum, refused to allow him to die because it would be the first time any Q had died... and it could throw their stable eternal existence into chaos.
As a paragon of Continuum living (reformed), Classic Q is dispatched to argue on the state's behalf, that Quinn should be forced to live forever in shrivelled soulless emptiness AND BE HAPPY ABOUT IT, DAMN IT! Judge Janeway runs an impromptu court so kangaroo-ish that it makes the 'Squire of Gothos' look fair and dignified. Tuvok is Quinn's advocate, as a Vulcan from a culture that saw the logic in suicide for those of advanced age and infirmity.
Q's case hinges on the infinite wonder of life, the universe, and everything, with character witnesses Isaac Newton, a Filthy Hippie, and William Riker brought in to illustrate how Quinn had improved humanity. Sparking the invention of human physics, helping wacky goofballs find true love, and saving the life of Riker's Civil War ancestor, Quinn's existence was pretty good for us human types.
Quinn's case is made on a field trip to a manifestation of the Continuum that mortals might understand: a rest stop on a road in a desert where the unimaginably bored Q dwell in silent tedium. Why silent? "Because it has all been said." Quinn manages to convey how bleak that really is: the clocks have no hands and the Q have nothing approaching feelings anymore. Only "sugar" and "drunk".
Janeway turns down Q's bribe to send the ship home, and rules in favour of Quinn's right to choose. She also asks him plaintively to at least try a normal mortal life. "I like this life. You might, too."
"Death Wish" is amazingly good if you need to heap more praise on a story about the laudable side of killing yourself, which I have more than mixed feelings about. First of all: don't. Second: I don't want anyone to suffer intolerably. Thirdly: I'd like to live for billions of years, myself. So, in conclusion, Man Up, push through the pain, and live forever. Like Star Trek! Star Trek refuses to die, even when it has all been said. Each of us should ask "Have I had enough?" My answer is no.