Julian delivering medical news in a crowded bar is very symbolic of his relationship to medical ethics.
Julian Bad Bitch Bashir knows full well that all Quark cares about is the malpractice money and Fuck all else. Plus Medical Ethics started the fight with Julian, so he’s allowed to be a little pissy with it.
The only reason there is drama in this episode is because Quark told Rom "the fuck does Bashir know?" Bashir would not have misdiagnosed Quark (because Bashir can't be bribed by Quark's political enemies. Come on, this whole episode was a set up by Brunt) and Bashir willingly escalates the drama because of Quark's snub.