Do you NEED to be on this quest to get the drop chance? Or does it drop as long as you've accepted the quest at least once?

I've got three more zeta weapons to exchange for relics, but I'm waiting to see which ones I want. Probably going to level a dps through fates before I make my decision, but I don't know if I should pick up the quest for a random class BEFORE starting my fate grind. As long as I'm grinding fates, I might as well be getting the crystals.