Gibrillont asks you to deliver the wine to a resident of the Pillars who is particularly fond of the blend.
This quest is available if you have completed the quest The Spice of Life.
Starting the Quest
Talk to Gibrillont in Ishgard to begin the quest.
Deliver the Twice-Mulled Wine
Go outside and use an Aethernet Shard to go to the Athenaeum Astrologicum Aethernet Shard. Go northeast to where the orange arrow is pointing in the minimap and talk to Hierytha to hand over the Twice-mulled Wine.
Report to Gibrillont
Use an Aethernet Shard to go to the Forgotten Knight Aethernet Shard (or go east from Hierytha to enter the Brume, then go up the stairs to the back entrance of the Forgotten Knight) and go to the quest icon on the map to talk to Gibrillont and complete the quest.
Next Main Scenario Quest
After you complete the quest, talk to Gibrillont to accept the next quest: A Child Apart.