For anyone interested, Tarantino spoke ad lib about the idea of making a Trek film back in 2015. I mentioned this in another comment here but didn’t have the link to the interview.
Go to 3:47 for the relevant section. Interestingly, rewatching it, the prompt of the conversation was “what Star Wars movie would you like to do” and Tarantino responds with he’d rather do a Trek film.
He’s main point is that so many good episodes from Trek, especially the original series, could be made into movies, and cites specifically City on the Edge of Forever and Yesterday’s Enterprise, which certainly indicate that he has some good Trek Taste.