From wiktionary:

>Adjective
>nostalgic
> 1. Of, having, or relating to nostalgia.
> 2. Reminiscent of the speaker's childhood or younger years.

>Noun
>nostalgia (countable and uncountable, plural nostalgias)
> 1. A longing for home or familiar surroundings; homesickness.
> 2. A bittersweet yearning for the things of the past.

Based on 2 of the adjective form, it would be correct to say "This music is nostalgic" meaning "This music is reminiscent of my younger years". However, it would also be correct to say (using form 1) "This music makes me nostalgic", meaning "This music makes me have nostalgia".

I have not, in my memory, heard/seen 2 used in actual conversation/writing. The construction "nostalgic for X" is far more common. I suspect this is because the implied specificity to the speaker makes that definition of "nostalgic" awkward to use.