After you retrieve the Nashacite at Drunken Stag's request, he passes along some information.
This quest is only available if you started the game in Ul'dah, you have completed the quest Give It to Me Raw, and have reached level 11.
Starting the Quest
Talk to Drunken Stag east of Horizon in Western Thanalan to start the quest.
Slay Sun Midge Swarms
There is an orange circle nearby on the map and minimap. Your minimap will display a small quest icon at each location where a Sun Midge Swarm can be found. Slay four of them to complete this quest objective.
Report to Fufulupa at Horizon
Afterward, go west to Horizon and talk to Fufulupa to complete the quest.
Next Main Scenario Quest
After completing the quest, talk to Fufulupa to accept the next quest: Last Letter to Lost Hope.