I'd say the proper adjective is a Japanese garden
A proper adjective is an adjective formed from a proper noun. "Arabian", "Grecian", "Japanese", "African".
The proper adjective for southwest is "southwestern."
The proper adjective is Haitian.
The proper adjective for Portuguese is "Portuguese."
Yes, if you are using it as a noun, i.e. to refer to a native of Japan, or the langauge spoken there, it is a proper noun. Otherwise it is an adjective.
Vietnamese is the proper adjective for Vietnam.
The proper adjective for Georgia is Georgian.
The proper adjective for George is Georgian.
Freudian is the proper adjective for Freud.
Mayan is itself a proper adjective.
The proper adjective for Siam is Siamese.