As you play the Accompany activities and form relationships by interacting with the people around you through daily visits and exchanges of flowers the Friendship will eventually reach the specific requirement needed to start a close relationship.
Although there are 5 types of intimacy in the game (Confidante, Bosom Friend, Lifelong Friends, Male Intimate and Acquaintance) there can only be one close relationship with the same friend. The most important of the 5 types of relationships is Confidante, which is the couple. This relationship is achieved with your friend after reaching 2000 Friendship Points, the couples rank can then be upgraded so the they receive generous rewards through daily tasks such as watering the Destiny Tree.
Note: When you delete a friend, the relationship will disappear along with the Friendship Point.