The fountain in the marketplace isn't working because something is blocking the mechanism. The Judge already sent two guardsman down into the catacombs below to check on the damage to the mechanism, but they haven't reported back. He asks Scarlett to go into the catacombs and take care of the matter.
From the the Judge's courtroom in the Outer City, go to the canal and down the steps of the nearest gondola station, which is near the central bridge. Just wade into the water and swim under the bridge (Scarlett automatically starts swimming when she walks into deep water, putting any weapon she had drawn away) Find an opening into the Outer City Catacombs under the bridge.
Swim to the nearest ramp, walk up on the platform (Scarlett automatically returns to walking when she goes up a ramp out of water), turn left and then right, and follow the tunnel to the waterwheel mechanism that powers the fountain. (Click on map to see a larger version.)
In the area just before the waterwheel, Scarlett will meet the two guardsmen the Judge mentioned, comically debating what to do. Scarlett can just proceed into the room, ( your journal is updated when you do so ), kill the Grippers, and remove the stone from the mechanism.
Then she should return to the Judge to complete the quest and claim her reward. Also, she should stop by the fountain and see what washed up their once the water began to flow again.
In addition, there is ample treasure lying around down in the catacombs. A girl's gotta eat.
When Scarlett returns to the Surface, she will see that the fountain is working again. She should go to the fountain in the Marketplace and check out the small jewelry box that somehow popped up there once the water began flowing again.
As a further bonus, not only the fountain, but several drinking fountains throughout the Outer City will begin to work again. Fresh water completely refreshes Scarlett's Mental Energy, which makes a working fountain convenient when she's out clearing Rogues or Marauders from the nighttime streets.