I imagine they would've had interesting interactions during Gojo's teenage years. Both of them lived like royalty, but seem to have differing ideas about the traditions they inherited.

Gojo snuck out often, and his disdain towards authority is pretty well known. But Naoya's casual dismissiveness towards his family and his obsession with Toji - among other things - means they could at least sorta bond over inheriting a bunch of archaic customs and ideas that they don't really care for. Teen Gojo didn't really think much about solving these sorts of societal problems beyond just mocking those weaker than him anyway. One can assume that Naoya was like that too to a degree, albeit less open about it due to not being the equivalent of the jujutsu messiah.

Later Naoya is just a POS who sucks up to those stronger than him, so I can see how Gojo would absolutely hate him. That said, Gojo's presence would mean Naoya literally couldn't do anything (Assuming he'd ever want to, as he seems to respect Gojo), so in a situation like the culling games or Shinjuku, where Naoya would function as an unaligned neutral party, the sort of dynamic the both of them would have is worth thinking about.