Does discarding crew count as my move?
No. Discarding active crew members are equivilant to playing an action and would still allow you to take a move this turn (assuming you have not taken a move this turn). Crew with discard effects from you hand (Like crew with the ability PICK or Bruno the cat) can be done at anytime as if it was an action, and does not count as a move.
Remember: You get one MOVE on your turn and it can be spent building (crew/hangar/armor or expert pilots) or attacking.