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.

Treasure is 보물 in Korea. Cherish is 소중히 여기다, 아끼다