Requirements to unlock secret dungeons:
1. Must complete all maps for said town (1-3 stars), secret dungeon wont appear unless you do so.
2. To get inside secret dungeons, you require a key. If in a party, all party members must each have one key.
To obtain secret dungeon keys, you would need to finish a repeatable quest. (this quest appears after doing all the quests for said town)
For the 2nd town, you can get the quest from this npc. The quest is to...
Got quests? Can't read chinese? Don't know where to do them?
Well, worry not since this guide could help you a bit on getting your way around questing in Elsword.
The image below shows my current quests. The first one is under the red tab, those under the red tab are usually event quests. Second one is in orange which is a quest in Besma, right next to it is the map on which the quests needs to be done.