Are you actually queueing for the dungeon as the same Job you started the quest as? Quests usually require you to be that class or Job you accepted the quest as in order for proof of completion for the quest to be rewarded (the npc the quest finishes at will usually have their quest icon in red with a key icon on it if this is the case until you change to the required class or Job).

Also, look at this page and check the steps required for the quest have indeed been followed as you may be missing a step somewhere in the quest before it will be listed as all quest objectives completed.