Traditional German (including other German-speaking countries) Christmas songs include world famous songs like "Silent Night, Holy Night"
or "O Christmas Tree".
Furthermore, there are also many songs that are not that popular as exports:
There are also less liturgical songs, but they are rare, the only one I can think of from the top of my head is this song, which is set to a tune that is supposedly of English origin, but without documented lyrics or something of that sort: