by demiurg
Not during the quest phase, no. You can only pick up rewards from a defeated hero marker during your "turn", and it's not considered any player's turn during the Quest phase (even when activating your heroes) in terms of game definitions.
I know that must be my turn, but can my hero take Reward cards, if he moves (Mobilize/Conquer card) through area containing a defeated hero marker (don't stop there)?