to cherish = 소중히 여기다, 소중히 간직하다, 소중히 기억하다.
treasure = 보물 (precious thing), 보배 (precious person or thing), 소중한 사람 (precious person), 소중한 것 (precious thing).
For both phrases, the adjective 소중하다 (precious, important, cherished) is the underlying idea. 보물 and 보배 seem to lack the sense of personal attachment.