expression, i-adjective
tight-lipped; able to keep a secret
Describes a person who does not reveal secrets or private matters. Often used in the negative form 口の堅くない to mean someone who cannot keep a secret.
See also: 口が堅い
彼は口の堅い人だから、秘密を話しても大丈夫だ。
He's a tight-lipped person, so it's safe to tell him a secret.
口の堅い同僚にだけ、その情報を伝えた。
I only shared that information with a colleague who can keep a secret.
Same meaning but uses the subject particle が, forming a clause rather than an adjectival phrase. 口の堅い is the attributive form modifying a noun.
Compound of 口 (mouth) + の (possessive particle) + 堅い (hard, firm). Literally 'firm of mouth', idiomatically meaning someone who keeps their mouth shut.