You can start this quest anytime after the events at Fisherman’s Horizon, after you have control of a mobile Balamb Garden. To start this quest you’ll need to go to Winhill, a small town along the southwestern end of the Galbadian continent.
Approach Winhill and note there are two entrances to this town - if you enter via the eastern entrance you’ll find yourself in the town square, while if you enter via the western entrance you’ll be near the item shop. Use the eastern entrance and head into the large house at the top of the town square screen, then talk to a rather… nondescript man wearing beige/brown who is standing on some stairs. He’ll whine about some heirloom vase and a ghost, which is all the prodding you’ll get for this quest.
Time to hunt down this vase! You should know the layout of Winhill well enough, so let’s laser-focus on what, exactly, needs to be done:
Make sure you have Quistis or Irvine in your party and search a suit of armor along the left side of the mansion. After some chatter, move away from it and a rather odd critter will flee from the dismantled armor, leaving a vase piece in its wake.
Exit the mansion and enter the lower of the two houses to the left - you should recognize it as Raine’s bar from Laguna’s dream - and head upstairs to get yelled at by a blonde woman. Talk to her until she mentions some flowers downstairs, then head back down and search a table along the lower-right of the screen with some white flowers on it. This should make an apparition of Raine appear behind the bar, which will turn into a cat when you talk to it. Interact with the cat to get a second vase piece. Either this town is haunted, or Ellone’s dream-sendings have messed with your mind in a bad way.
Return to the town square and exit via the bridge at the bottom, then in the next screen enter a house to the left (at a three-way intersection in the road) where an old lady lives with her flowers. Search some white flowers at the bottom of the screen. The old lady will ask if you like flowers - respond with Yes and she’ll babble about Raine, then search the same flowers again to find another vase piece.
Exit the old lady’s house and exit down to the next screen to find some chicobos running about. Standing in the intersection through which they run and interact with one to send it flying. Err… is that some animal cruelty or…? Whatever the case, it should drop the final vase piece. Chase down another chicobo and you’ll be rewarded with some Gysahl Greens, then two more to score a Phoenix Down. Nice. Abuse too many chicobos and they’ll become wary of you, fleeing at your approach… and a mama chocobo will get some vengeance! Yikes.
Return back to the man in the mansion and give him his broken vase to receive… a Holy Stone. That’s right. Just one. That makes this little vase quest about 1/10th as profitable as winning a Krysta Card would be. Oh well, at least it’s done.