The word 'quite' acts as an adverb. In this sentence the adjective 'young' is modified by 'quite' as 'young' itself has no extremes of meaning. 'quite' suggests that the person was not a child nor yet old-- adult, but still at the lower end of adulthood. In another sentence, 'quite' might reduce the force of an adjective, such as 'quite' angry, suggesting the person was not furious.