The relic quest is named "A relic reborn". Gerolt, an NPC in North Shroud (30,20), Hyrstmill, gives the quest.

It is a long quest that ends in you obtaining the Curtana (the relic weapon for the PLD.) You have to defeat enemies, do dungeons, trials, etc.

Later the Curtana can be upgraded to Curtana Zenith, and later on to Curtana Atma and Animus, and in the future more upgrades are expected. So it is worth to do the long quest and longer upgrades.