How to get this quest
You can find this quest in Trademond's guild after completing What Is It with Birds And Stealing and waiting a day if you are at Intermediate rank.
The northern mountain ranges
The northern mountain ranges are divided into three areas.
To reach the demon's domain, you can go follow a path going through an avian's nest or a shortest path going through a curse trap. There is also an ogre king walking around Lilim's domain. This enemy is a resilient and relies on strong physical skills, so tools like poison coating + Defensive Stance, Pacify, or inflicting the Numb or Blind state are recommended to defeat it without removing Aura's collar. With good timing you can avoid fighting the ogre king and head straight into the domain.
Fighting Lilim
Lilim's moveset contains Foggy Pollination (an party-wide ATK debuff) and Luxuria (inflicting Charm). Stacked with the area-induced blindness, this physical attacks close to worthless against her. Lilim's offensive moveset contains a normal attack hitting three times and Cupiditas (lowering your defenses). Lilim's ATK is not that high though, so a triple attack against a party member without debuff won't be too dangerous. Cupiditas also does pretty low damage unless the two negative energy orbs in her domain aren't dispelled.
The main issue is to damage Lilim efficiently to not lose an attrition battle. Lilim being able to heal herself makes it even worse. Lilim's MDEF is high, so spells won't hurt her much. If you don't want to remove Aura's collar, you need to debuff Lilim's MDEF and partner with the fire slime summon. For this, you can use an acid bomb or use Flaming Robe to exploit Lilim's triple attack to your advantage. Once Lilim's MDEF is debuffed, you can hit her with Fire or Light (if at least rank II) to do big damage.
Once Lilim's flees the battle, you can report to the guild clerk to complete the quest.