Though it takes place in one of the earliest sections of Chained Echoes, the side quest A Goblin's Dilemma won't be available to you until fairly late in the game. To unlock this quest you must first complete Ruins Under a Tree, which involves traveling tot he Empyrean Ruins and completing its many puzzles. Once you've done some cooking with Amalia back on Hermit's Isle and regained control of Glenn you can begin this quest.
Travel to the Rohlan Fields and head to the northern region. At the top of the waterfalls that descend into the lake in the middle of the Rohlan Fields is a tree, and tied to that tree is a goblin. His name is Bark, and he'll insist that he is most certainly not a goblin. Stuttering wildly once freed, he'll run off to Basil to explain his situation to the people who tied him up in the first place.
Follow Bark to Basil. You won't find him, but if you look one screen east of Basil you'll discover a villager inspecting some vicious claw marks in the ground. There were plenty of indications of a monster running loose in the Rohlan Fields earlier in the game, and it appears that you're finally about to receive some answers. Lenne will mention Jan, the scam artist from much earlier in the game, whose home was attacked by the beast.
Fly to the Seaside Village and visit Jan's home, on the east side of the village. His daughter Marsia is peeking through the window when you arrive, and when you go inside you'll find the people of Basil blaming Jan for the attacks. Things will move quickly, and you'll watch a squad of goblins rush away from the scene outside. The village elder will give you a Goblin's Crest and ask you to mediate the situation for them.
Your next destination is the Goblin Village, deep within the Ograne Grottos. If you haven't already visited this section of the grottos you'll need to use the northern entrance in the Rohlan Fields. You'll find it at the bottom of the waterfalls from earlier, where Bark was tied up. This guide will take you through the Ograne Grottos if you need directions. You'll also need to complete part of the side quest The Gates of Nhysa to reach the Goblin Village, as there's a nigh-unbeatable monster in your way otherwise.
Head down to the central Fast Travel Crystal in the Ograne Grottos, walk north, then take the first path on your right to the north. This leads to the Goblin Village, and with the Goblin Crest on your person you'll be allowed past the sentry. This... eccentric... place has a few things to see and grab before you proceed with the mission:
- In the northwest of the village you'll find a merchant. He'll only sell you three items, but they're all important. First are Incocybin Berries, which make the Tadeyes of the Ograne Grottos beatable. Second is The Human Language, which allows you to understand the language of the goblins. Last is Baibai X, an item that will grant you access to additional areas in the Flower Fields of Perpetua.
- If you bought The Human Language you'll learn from a goblin in the northeast that he dropped a key in the mushroom field in the northwest, near the merchant. Check the sparkle that appears for the Miner's Key.
- In the northeast you'll find another mushroom field. If you walk west from the mushrooms, behind some tents, you'll find a red chest containing a Hall.
- South of the northeastern mushroom patch is a southbound path that appears to lead nowhere. You can follow a hidden path west of here to find a red chest containing a Magic Poleyns.
- If you tell Egyl to kill the goblin the Goblin Chief will show up, reveal the truth behind things, and order his goblins to attack. You'll need to battle three Goblin Knights and two Goblin Wizards. Jan will declare that he never wants to see the party again.
- The scenario will play out much the same if you choose to knock the goblin out, and you'll again need to fight three Goblin Knights and two Goblin Wizards. After the battle, though, Jan will take in the goblin, and you can see them in a secret little cave behind his home.
The 'good' ending is sparing the goblin, and you'll see more story if you go this route. Regardless, the quest ends when you drive the goblins away... though it's technically not the end of the goblin problem.
Head to Rockbottom in the Kortara Mountain Range. If you get down on the mine cart tracks and head north you'll find an elevator down to the Wygrand Mines. Head east from the bottom of the elevator and you'll find a door that was, previously, blocked by a goblin. Assuming you picked up the Miner's Key from the Goblin Village earlier, you'll be able to go through the door.
The passage beyond the door will take you past several goblins (and battles with goblins) to another elevator. At the bottom you'll find a small altar with a Darkness Tablet that you should absolutely grab, as well as a path to the goblins' shrine of worship. You'll have to jump over to the totem they're worshipping, and you have the option to set it aflame. Doing this will essentially wipe the goblins out of the game.
Go back to Jan's home near the Seaside Village. If you decided to spare the goblin you'll find Jan in the cave behind the house, and, overjoyed by how things turned out, he'll give you an Elixir, two Repair Kits DX, and two Platinum Ores. All's well that ends well.
(One final note: If you go back to the Goblin Village after burning the totem, the merchant selling items will still be there. He'll just look a little different. Nonetheless, you don't have to buy everything he has before completing the quest.)