I don't think that's possible.
The NPC that gives the quest always appears in that dungeon after you kill the fairy. If you just enter it, then yes. You will NOT see the NPC there. You need to kill the second boss in solo mode for that NPC to appear. For the second boss to appear, you need to kill the first boss.
If it will appear and will not have a quest, then you may have a quest in the forest that will take you to the dungeon again. If not, then you will have to send a ticket, probably.
The sub-class quest is from Andre, but the one that sends you to him is archbishop. Before that, you need to finish the dungeon and someone will send you to that archbishop. Just finish that series of quests and you will be able to enter demarech mines.