Is it possible to pass on Event exclusive moves through breeding?

Solution 1:

It is not possible to get Surfing or Flying Pikachu in Generation 6 games, because Pokémon that know HM moves cannot pass through transfer systems.

Consequently, if someone is offering you such a Pikachu in X and Y, you can be pretty sure it's hacked - at least, until an event is run distributing it!

EDIT: Upon a little more investigation, even if a Surfing Pikachu event is released, breeding it will NOT give another Surfing Pikachu, because Pikachu is not compatible with the Surf HM and it therefore will not be passed down - only TMs that can be learned by the child (in this case, Pichu or Pikachu as the case may be) are passed.