This quest is available starting Chapter 7.
Go to the Market through the south end. There will be a small cutscene for you to watch. Talk to her 3 times, but not all at once, complete chapters in the meantime and try to get a date with her before Chapter 22. There will be a cutscene later on, and she will show you her cat. From time to time, make sure you give the cat almonds. After you become a knight in Chapter 30, she will give you your reward.
Reward: Dragon Fang