After Teteroon finishes brewing the firewater, he asks you to deliver it to Buscarron.
This quest is available if you have completed the quest Microbrewing.
Starting the Quest
Talk to Teteroon in the western half of Upper La Noscea to begin the quest.
Deliver the Qiqirn Firewater
Go back to the South Shroud and talk to Buscarron. Hand over the firewater.
Next Main Scenario Quest
After you complete the quest, talk to Buscarron to accept the next quest: Sylphish Concerns.