This quest is available in the Adventurers' guild a soon as you join.
The quest takes you to the Northern Mines. In order to enter, you need to either use Star Knightess or a bomb. You can buy a bomb or you can find one in the Forest of Runes.
If you talk to Charlotte outside the mine, you learn that an experienced adventurer could teach you a way to fight the spider queen more easily. After talking to Charlotte, go talk to Jacob. You'll learn the boss's weakness.
Alternatively, you can learn the spider queen's weakness from a book in the monstrologist's house.
Note: there is a body just to the right of the mine entrance. The body can be looted for 25g. This body will disappear after completing the quest.
Once you're inside the mine, it is advised to go around the area counter-clockwise. You'll find a standing enemy on the way. Fighting this foe leads to a shortcut after a battle which can be cleared without taking any damage (only 3 spiderlings are fought).
The room where the spider queen is is to the left of the entrance, but the direct path is blocked by a door. The door can only be opened from the other side. It is possible to destroy the door , but doing so lead to a reduced monetary reward.
There are four torches close to the boss. Charlotte can cast fire on each torch to light it. If you light all four before fighting the spider queen, it will have the Defenses Broken debuff, reducing it's DEF and MDEF by 75%. In this case, normal attacks are good enough to hurt the queen. Else, Pierce I is the only good damage option if the spider queen is not in the Enhanced state or the party's ATK isn't buffed.
In the fight Charlotte should, since the queen is resistant to fire, buff the the party's ATK. John should use Protect I if Charlotte is in danger and didn't fully buff the team yet. Aura should guard or attack. Paul should attack (broken defenses) or use Pierce I (not broken). When the queen is low on health she summons four spiderlings. John can kill them all with Slash I.
Go to the guild and talk to the clerk to complete the quest.