JJ has told the story that his first exposure to ST was when his Dad took him, as his +1, to the gala world premiere of ST:TMP at the Smithsonian Institute. JJ was just a kid and he was bored silly by the movie only a few years after loving "Star Wars".
I was 21 when TMP premiered, and I knew random episodes of TOS and TAS. I was at the right stage in my life to fall for the whole ST phenomenon. I met many TOS fans who hated TMP, and many SW fans who preferred SW to ST.
In recent interviews, JJ gushes about the TOS-related characters. He fell in love with them while making the 2009 film.
Does he need to know any more than that to direct a ST movie? Certainly he knows more than Stuart Baird did about TNG.