You can only start this quest if you are on friendly terms (or rather, just not hostile) with the Mites in the Old Sycamore area, which you can do when you first encounter the warring factions in the exterior area upon first entering the Old Sycamore - just pick the [neutral] option, or actively side with the Mites. Once done, enter the Mite Caves sections of the Old Sycamore Caves and, after fighting through some Giant Centipedes, talk to Gatekeeper Quoggy.
Pick the dialogue options “What lies ahead?” followed by “How do I get to this broken passage?” to start the quest Mandible Jewelry. Apparently the Kobolds and Mites both sealed off a communal passage when hostilities started, leaving some treasure behind in the process. If you acquire a particular giant bug’s mandible and bring it back to Gatekeeper Quoggy she’ll give you a lever to allow you to lower the bridge leading to this treasure-filled no-man’s-land.
Make your way through the Mite Cave and use an area transition southwest of Queen Bdaah to reach the Old Sycamore Depths area, where you should find yourself along the northeastern edge of the Old Sycamore Depths. From where you start, head south to find more Giant Centipedes, this time joined by a new variant - the Spitting Giant Centipede. After the fight with the centipedes, the path will fork. Ignore the branch to the southwest and continue south until you find… another fork! This time take the fork to the east and follow it until you reach a watery chamber guarded by two Giant Centipedes. Smite them, then turn your attention to the… ah… giant bug in the northwestern corner of the chamber, which you can search for the Mandibles of a Huge Bug.
With the Mandibles of a Huge Bug in your hands, it’s time to leave the Old Sycamore Depths for a bit and finish up the caves above. Depending on your past actions, you can return to Gatekeeper Quoggy and claim the Bridge Hoist Part, should you not have claimed the device earlier through violence. However you managed it, now that everybody has this Bridge Hoist Part you can finally finish off the Old Sycamore Caves area and claim the treasure in the no-man’s-land. From Gatekeeper Quoggy, head south to the cavern in which the Mites are (or were) sorting junk, and from there take a passage in the southwestern corner of the chamber, which leads south to another chamber full of Mites.
Do what you will regarding the Mites, then head south to find the raised drawbridge… a rather more elegant solution than the Kobolds had, truth be told. If you have the Bridge Hoist Part interact with the bridge, then cross to the south to reach a chamber which clearly saw a great battle in the recent past. Loot some rocks to the west to obtain a Scorched Fragment of a Necklace, then loot a chest along the eastern end of the cavern to find some trinkets and a Greatsword +1.