I guess it depends on what you mean by *believe.*

I belong to a folk music group, and at our December meeting, many sing holiday songs (as you might well imagine). We have a little 'open stage' segment, where we can perform songs for each other, and we often collaborate on performances.
I have joined with some of my Jewish friends to sing songs like "Light One Candle," and "Rock of Ages" (which is different from "Rock of Ages, cleft for me..."), although I am not Jewish. Likewise, some of them have joined me on Christmas songs, although they are not Christian. No one has ever batted an eye.

You can join in a song and be believeable, I think, if you have *respect* for the tradition from which it hails. This does not extend only to religious songs, but to any song that comes from a culture not your own.