Probably not. Tolkien didn't own a television and wasn't really interested in modern culture, preferring to attend the opera, lectures on poetry, and listen to occasional programs on the wireless. He certainly watched some TV (when visiting his brother's house he noted that they had watched the cricket and tennis while drinking whisky) and while there's no evidence that he didn't watch Star Trek on someone else's set, it seems unlikely.
He never spent money carelessly; he and Edith did not install any
electrical gadgets in the home, for they had never been accustomed to
them and did not imagine that they needed them now. Not only was there
no television in the house, but no washing-machine or dishwasher
JRR Tolkien: A Biography
He described the radio in extremely disparaging terms.
Only in one way was I better off: wireless was not invented. I daresay it had some potential for good, but it has in fact in the main become a weapon for the fool, the savage, and the villain to afflict the minority with, and to destroy thought. Listening in has killed listening.
Tolkien Letter 61 (to his son, Christopher)
I think we can be reasonably sure that if he was even aware of Star Trek (noting that when introduced to film megastar Ava Gardner at the height of her fame, he hadn't heard of her, let alone seen one of her films), that he would have had little or no interest in watching it.