Claire is Emily Young's niece. When she visits La Push, Quil imprints on her--creating a scandal among the La Push werewolves because Claire is only 2 year old at the time. However," there's nothing romantic about it at all, not for Quil, not now (Eclipse, pg. 176).
Claire's only actual appearance in the series is in Breaking Dawn. Jacob finds Quil and Claire playing at First Beach and he tries to convince Quil that it would be all right for him to date while he waits for Claire to grow up.