In recent times, yes. But in 1989, not really.I'm willing to bet more people were reading Dr Spock, than watching Mr Spock.
But that's just it: Other than Benjamin Spock, have you ever heard of anyone else on Earth with the name Spock? I've heard of tons of Caitlyns (or Katelyn, or Catelynn, or however you want to spell the name that sounds the same).
By the association with Benjamin Spock, most people would recognizes it as a human name. Caitlyn sounds made up or a combination of Kate and Lynn. Looking at a name site, it was pretty much non-existent prior to the 1970s.