This 'quest' is given through the Vault-Tec Population Management System, which allows, apart from other things, to track companions by marking them as quest objectives. Once a marker is activated for a companion, this quest will begin, ending once they are found. This is particularly helpful for tracking down companions 'misplaced' during travels. More than one marker can be active at a time.
PC Unless the player character manually deselects the quest in the Pip-Boy, when they attempt to "shut off the tracker" it will remain active, and not allow new quests to be selected automatically.[verified]