There's no connection at all. The city's name (not nickname) comes from a Celtic-speaking Gaulish group that lived in the area, the Parisii; the mythological Paris gets his name from a Bronze Age Luvian word, pariya- 'outstanding', which is also the root of his father's name, Priam.

There's no reason to suspect any link. Shared names, and similar names, are often just a consequence of the fact that there are lots and lots of names in the world, and only a finite set of phonemes to go round. Sometimes you get chance resemblances. The name also appears on the herb paris, and that has nothing to do with either of them (the herb comes from Latin par 'equal'). Similarly Winchester has nothing to do with chests, and Easter has nothing to do with Ishtar.

(Conversely, when names are related, it's often disguised: like 'Priam' ~ 'Paris', or 'Winchester' ~ Latin castra. Not to mention general vocabulary like English 'church' ~ Greek kyriak-, 'five' ~ Welsh pump ~ Spanish cinco, or 'calf' ~ Greek adelphos. But to go over that thoroughly you need loads of training in comparative linguistics.)

Edit - made a booboo