The quilt is a very important aspect to the main questline of the game Harvest Moon: Tree of Tranquility. With it, you can begin to call upon the five Harvest Sprites and complete the rainbows, which will unlock new characters and areas.
There are two halves to the quilt, and the mayor will speak to you when it's time for you to learn of it and begin the main quest.
Learning of the Quilt
To even learn of the quilt, you must first ship 1000 G worth of goods (this can be easily accomplished early in the game by shipping mushrooms found in the mine and planting crops). When you have done this, Mayor Hamilton will appear at your house the next morning, and comment on how well your farm is doing. He will say it's time for you to learn of the quilt, and a short scene will play out as he tells you a few basic facts about the quilt.
First Half of the Quilt
You will get another visit from Mayor Hamilton when you ship another 1000 G worth of goods. Again, he will show up in the morning and comment on your farm, then you will automatically go to his house to discuss the quilt. He will then present you with the left half of the quilt, and tell you that you must find the other half. In his dialogue, he will hint that Gill is the last person the Harvest Sprites have spoken to, so he is the next person you need to speak with.
Second Half of the Quilt
The second half of the quilt requires a little more work to obtain. Speak to Gill, and, if you haven't already upgraded your house once, he will ask you to do so (reason being he thinks that you're "just settling in" and have different things to worry about than the quilt). Once you've upgraded your house, he will direct you to speak with the carpenter, Dale. After some dialogue, Dale will direct you to Mayor Hamilton, telling you to tell him that you're interested in fixing the clock tower. He will then give you the key to it.
With the key, head to Waffle Town Square, where the tower is located, and equip it like a tool. Face the front of the clock tower and use it. This should trigger a scene where you retrieve the second half of the quilt, and with it, you can now begin to reawaken the Harvest Sprites and complete the rainbows.