Mogmug asks you to defeat some of the archeosaurs that have been too overbearing in their adoration of the moogle.
This quest is available if you have completed the quest Mountaintop Diplomacy.
Starting the Quest
Talk to Mogmug in Moghome in the Churning Mists to begin the quest.
Go west to the slope that leads to the western archway out of Moghome. Go up the slope and through the archway, then go north from the Aetheryte and continue north to the orange circle on the map. Dismount and enter the purple circle. Defeat the two Irate Archaeosaurs, then defeat the Bull Archaeosaur that appears.
Report to Mogmug
Go south to the Aetheryte and go through the arch between the two fluffy white lights. Go to the quest icon on the map and talk to Mogmug to complete the quest.