This quest has multiple solutions. In order to trigger it, you must talk to the owner of the stolen painting.
Objective:
She lives near the entrance to Tarant, pass the first bridge after the Boil, pass the first house row and then turn left. The Garringsburgs live in the middle house. Inside, you will find Evelyn Garringsburg, who'll tell you about her painting "Kerghan and Pershephone" was stolen from her. You can offer her your service in finding the painting. She will tell you to read the newspaper which is in on the floor, in it you'll learn that her husband is a member of the Wellington Gentlemen's Club, and she'll also mention that her husband might've talked about the painting one evening while drunk.
Locations, Clues & Walkthrough:
Go to the club, you can ask the doorman for a copy of the guest list (you'll have to bribe him for it). The only non-member in the club that evening was Rorry Limes. Go inside the club and talk to the Dark Elf Noble in the rear corner. He will give you a note that you're supposed to deliver to 36 Low Dervish Road (in the industrial part of Tarant), but he will also tell you not to read it.
You now have two clues to go with. Go to the Hall of Records and look up Rorry Limes. You'll learn he has no home in Tarant, but he owns a building in the industrial area, at 57 Mulligan Bone Alley. If you read the note the Dark Elf gave you, you'll find that it also says 57 Mulligan Bone Alley. If you read the note before delivering it to 36 Low Derwish Road, you'll be attacked by the people inside the building. If you did not read the note, you'll receive 100 gold and 1000 experience, but you'll also lose some alignment. If you kill the people inside the building, you'll find that one of them has a passport with the name Rorry Limes.
Now, go to 57 Mulligan Bone Alley. Inside the building, you'll find a half-ogre who has a key. You can kill him or pickpocket him for it, or if you're persuasive you can say that you're working for Rorry Limes and have come to pick up the painting. No matter which one you choose, the painting is in a chest in the back room.
Rewards:
Return to Evelyn Garringsburg and give her the painting. You'll receive 300 gold, 1200 experience and some alignment. Also, the newspapers will feature you on the main page, on the article "OUTSIDER FINDS STOLEN PAINTING".
Alternate solution:
It's also possible to get Evelyn Garrinsburg to double up the reward. To do this, go straight to 57 Mulligan Bone Alley before triggering the quest at the Garrinsburg residence, kill the half-ogre guard and get the painting, and then return it to Evelyn Garrinsburg. Don't ask her to introduce herself, just tell her that you brought her painting. Initially she'll be slightly confused, and mention that you're probably interested in the reward. If you then choose the surprised dialogue ("Reward? Are you offering one?...") she'll get suspicious and accuse you of the theft of the painting. Then choose the dialogue to just give the painting for free ("Listen. This is no longer worth the trouble..."), and she'll get so embarrassed that she'll pay double. The downside is that you'll miss out on the fun detective part of the quest, but it really showcases the developers' foresight and the myriad of ways to approach quests. Done with 9 CHA and 1 rank + apprentice in persuasion.
Third solution:
If you are a member of the Thieves Underground, you can also sell the painting to Thaddeus Mynor for 300 gold, but no experience.
Fourth solution
If you side with Madame Toussaude against Dolores Beeston in the Fortune-Tellers quest, then Madame Toussaude offers to give you information about the Garringsburg Robbery. If you haven't heard of it yet, she says "Oh, you will...". On completing the quest then Toussaude uses magical means to tell you the painting is at 57 Mulligan Bone Alley, again even if you haven't encountered Mrs Garringsburg yet. You can skip all the detective work, though you still have to deal with the Ogre.