What Dehner said to Kirk was, "Gary told me that you've been friends since he joined the service, that you asked for him aboard your first command." I think that would be a strange choice of words if she were referring to the ship Kirk was currently commanding; in that case she would've been far more likely to say "you asked for him aboard this ship." And the whole point of her statement is to call attention to how long Kirk and Mitchell have been close, so it's unlikely that "first command" is recent. But we know Kirk can't have been in command of the ship for too long prior to "Where No Man," given what we know about how long Spock served under Pike and other timeline factors.
So IMHO it's very, very improbable that Dehner is referring to the Enterprise
itself when she says "your first command." She just would not have chosen her words that way if that had been her intention. By far the most probable interpretation is that, yes, Kirk commanded another ship before the Enterprise