Examine the Carrier Pigeon Letter permanent key item that you received during The Stars of Ifrit quest.
Talk to Carmelo who is at (E-6) in Port Bastok. He will give you Carmelo's Song Sheet and ask you to place it at the Mirror Pond.
Go to the pond at (J-7) in Beaucedine Glacier.
A portion of the pond will become selectable (the target is named "Mirror Pond") in the presence of dark-element weather or clear weather. This target disappears when there is single or double ice-element weather.
Select the "Mirror Pond" for a cutscene.
Return to Bastok and talk to Carmelo to complete the quest and receive your reward.
Carmelo (Steaming Sheep Restaurant, Port Bastok)
Carmelo wants to start his life anew. You must go throw the song sheet he had written for his former love into the Mirror Lake on the Beaucedine Glacier.