Firstly, for your sanity's sake, if you at any point intend to do the level 80 relic weapons then you want to have the first upgrade quest active while you are farming Heavensward FATEs, because those are one of the two ways you can earn the 60 memory crystals needed per weapon.

Secondly, are you definitely at the right stage of the Anima quest itself? The elemental crystals won't drop if the quest isn't active.