It isn't quite a quirk, but more like a specific aspect of himself: Lex is proud of his heel/rival persona.

So in Carst (I swear I did this before Gen 8 happened lol) league challenges are highly televised with rivalries and sponsorships and shit. Its really treated like a sport, and there are financial incentives for trainers to craft a brand for themselves.

Lex became an extremely self-confident, boastful rival sort of dude with the skill to back up his ego. Beneath it he's actually just very dilligent and studious, but plays it off like natural talent. Even though he's a polarizing figure he's very popular and well known as a result, and he couldn't be prouder.