Final Fantasy VIII Guides and Walkthroughs
Once you gain control of the Balamb Garden during Disc 2 you can revisit Winhill (the place you saw during the Laguna flashback at the start of Disc 2). You can find the small town of Winhill at the southern end of the western continent, use my screenshot above for help locating it.
Enter Winhill from the northern end if you'd like to be by the mansion which is where this quest begins. If you've forgotten what Winhill looks like already, the mansion is pictured below. The man we need to speak to in order to begin this quest is found on the stairs inside. Just go through his dialogue and when he stops giving you new dialogue you're good to go.
There are a total of four Vase Pieces that you need to collect for this quest and most of the loot you get from this quest occurs during the final stage of it instead of after completing it. I've listed all four locations of the Vase Pieces below and if you still need help finding them scroll down a bit further to the rest of the page.
1. Examine the suit of armor in the mansion with Quistis or Irvine in your party then try to leave. The Vase Piece will be on the ground after the scene
2. Go into the pub and speak to the women upstairs until she mentions flowers downstairs. Go downstairs and examine the flowers in the bottom right corner of the screen. Afterward speak to the woman behind the bar then examine the cat until you get the vase piece
3. Examine the flowers at the flower shop in the lower-right corner twice
4. Where the Chicobo crosses the road and the music changes you'll want to kick it three times. The first time you get the Vase Piece and the other two times you'll get Gysahl Greens and Phoenix Pinion
During the Chicobo part you don't need to press X or anything, just run through the Chicobo at the right time. The third item you get, Phoenix Pinion is mislabeled as a Phoenix Down in the game. With all four pieces of the vase in hand return to the person standing on the stairs and give them to him. In return you'll receive a Holy Stone.
Vase Piece in the Armor.
Vase Piece inside of the bar.
Vase Piece inside of a Flower Shop.
Vase Piece for kicking a Chicobo.