There is not a universal Messages setting that adds a separate password to individual text conversations. On supported iOS versions, however, you can lock or hide the Messages app itself using Face ID, Touch ID, or the device passcode.
- Touch and hold the Messages app icon on the Home Screen.
- Choose the option to require Face ID, Touch ID, or the device passcode if it is available on your iPhone.
- Authenticate to confirm.
- Open Messages again to verify that authentication is required.
For additional privacy, change notification previews so incoming message text does not appear on the Lock Screen. Remember that anyone who knows the device passcode may still be able to access protected apps depending on the feature and security state.
If your iPhone does not offer app locking, update iOS and check whether the feature is supported rather than installing an unknown “message locker” that asks for access to private conversations.