No, they cannot. I've clarified this to someone else before, quite a few pages ago, so I guess it's due for reiteration: Unless the GM(s) specify that they have a TM or Egg Move, or a Hidden Ability, you're restricted to moves learned by level and the base ability/ies. However, you can PM or VM Vato and/or myself prior to making a post and work out Pokémon moves.
Level is entirely up to the GM, but you can use any learned-by-level moves within the proper range prior to capture.
Basically, if you want to be special, tell us first. Generally, though, making an enjoyable battle is the better, and more rewarding, road to getting a Pokémon with awesome stuff.