If Larvesta (Pre-evolution) levels up with an Exp. Share to level 100 in a battle and learn Flare Blitz won't it evolve after the battle to Volcarona and then learn Quiver Dance?
yes if you leveled Larvesta up to level 100 you get Flare blitz and after it evolve you will have the option to learn fiery dance which is it's signature move. In order to learn Quiver dance and Volcarona's other level up moves you will need to relearn them with the heart scale. I do not know why you want flare blitz on a Volcarona as it is physical move and Volcarona's attack is aweful and it's S.attack is amazing. I would flamethrower or Fiery dance instead.
If you level it to lvl 100 it will first learn the attack during the battle and then evolve, just like Boronicolas said. But after it evolves there are moves that cannot be learned again.