During the game, as you speak with Peatrice at the Item Check more and more, she will start to become happy and flirty with you. Basically, you’ll have to keep speaking with her until she calls you “darling” with the text in red.
Eventually, once she does call you “darling”, sleep until nighttime and go to her house. It is found to the east of the Bazaar. Inside, note 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 bubbles, 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, sleep until the day or (preferably) night and then speak to Peatrice’s father to receive the Gratitude Crystals from him instead.