Termini Cliff is a quest available after the quests of Edena Village have been completed. It can be done once the player has reached Lvl. 28 and unlocked Termini Cliff dungeon, although it may appear in the quest list before the dungeon is unlocked.
Termini Cliff is the last dungeon of Murken Forest, therefore the map clearing bonus is awarded after passing the dungeon once and thus completing the quest.
To complete the quest, Termini Cliff needs to be passed, that means the end boss needs to be defeated.
Click on [Campaign] -> Select [Termini Cliff] -> Gain passage through the dungeon A dark and forbidden place, Termini Cliff is located at the far end of Murken Forest.