During the game, as you speak with Peatrice at the Item Check more and more, she’ll start to flirt with you. Basically, you’ll have to keep speaking with her until she calls you “darling” with the text in red.
Eventually, once Peatrice calls you “darling”, sleep until nighttime and go to her house (or go to her house and sleep in one of the beds there). Her house is located to the east of the Bazaar. Inside, check whether Peatrice’s father has a speech bubble above his head; if not, speak with him, exit, and re-enter.
Once there’s a speech bubble, speak with him and promise to help him. Next, sleep until daytime and visit Peatrice at her usual place. She’ll invite you to her house at night. Again, find a bed to sleep in (can be inside her house) until the promised time.
When you next speak to Peatrice, she’ll ask Link some very important questions. If you choose to accept her love, you’ll get five Gratitude Crystals from her straightaway.
Otherwise, if you reject her (Link, you heartbreaker!), sleep until the day or (preferably) night and then speak to Peatrice’s father to receive the Gratitude Crystals from him instead.