Listen, in that cartoon Brian did not vomit because he discovered that Quagmire's dad was a transwoman. He vomited because he had been fucking Quagmire's dad, the transwoman. And then he refused to tell anyone about it because he was ashamed, and showered compulsively but still felt dirty and violated, and couldn't have any sexual feelings without intense shame and depression.

You watch the show. Do you think that the character of Brian, written consistently as a fairly comfortable middle-class liberal intellectual, would have vomited if he'd found out that Quagmire's dad was trans when he met her at the bar? Do you think he'd have showered compulsively if he'd been hanging out with with Quagmire's dad at a party and found out she was born a man? I would likely say, not.

Again, I submit that Brian's reaction was not to the simple fact that Quagmire's dad was trans, but rather because his sexual identity had been seriously violated and as a result he'd been psychologically traumatized.