Yes you can, BUT you won't be able to change your job since it requires one class to be at 30 and other one at 15
for example if you want to play as scholar, you'll need to get to lv30arcanist and lv15 conjurer.
about the quest exp, you won't have to worry about it much since you can just level up your 2nd class with F.A.T.E./Dungeons since you'll gain additional 50% bonus for most thing you do.