If the quest is active, a character opening the chest will obtain the appropriate key item from the chests. Keys are needed to open the chests in those zones by all jobs except Thief, who can pick the chest.
Once all three key items are obtained, return to Bastok and talk to Carmelo for a short cut-scene.
Zone out of the area then back in, then speak to him again to complete the quest and receive your reward.
To help save Dariah from her strange illness, you must gather the pages of the "Chanson de Liberte." The missing parts -- "Un Moment," "L'Ephemere," and "L'Ancienne" -- lie scattered about in various locations around Vana'diel.