There is some crossover between Spock and Kirk at the Academy. We know Spock started the Academy eighteen years prior to "Journey To Babel". We know Kirk was 34 in "The Deadly Years" so if he started the Academy at 18 it was sixteen years prior to that episode. Its possible they met and depending on the way classes are structured, they might have shared a class or two. On the other hand they could have passed each other in the hall and never noticed.