You're doing the wrong Hunting Log - as the npc says you have to complete the second rank for the actual Grand Company Hunting Log and not the regular class Hunting Logs (selectable on the Hunting Logs window by simply clicking on the logo of your GC listed under the class icons, in your case, the Ul'dah logo).

Unlike the class Hunting Logs, the GC Hunting Log only has three ranks, with the second rank requiring you to defeat enemies that are only found in the Sunken Temple of Qarn, so you will have to queue for that dungeon to complete it (the third rank likewise requires enemies in Wanderer's Palace). GC Hunting Logs also only give you GC seals for completion of objectives instead of exp and gil.