As a Nuzlocke run is entirely self-imposed, nothing prevents you from doing a Nuzlocke variant when you have already completed the game. The natural endpoint would be beating the Elite Four with your Nuzlocke team.
Pick an arbitrary Pokemon to act as your starter. You can hatch an egg, and use exp share or rare candy to get to level 5.
Remove all other Pokemon from your party and return to the starting town.
Now just follow your preferred Nuzlocke rules. (only catch first Pokemon on route, release any faints, etc.)
Continue your run until all your Pokemon faint, or you reach your end goal (Beat Elite Four, Battle Tree, Full level 50 party, etc.)
You won't have to fight any Gym Leaders, but you can always fight wild Pokemon, which you do heavily in a Nuzlocke anyways to prevent losing any Pokemon.
I would recommend having a designated Nuzlocke PC box to keep track of which Pokemon you can use on the given run.