Go through the logfiles and check what errors you get. There has been some fiddeling around with nginx, so there is a version in the documentation, but not for lighttpd. sabredav seems to support lighttpd (Webservers - sabre/dav), and Nextcloud uses this (or a modified form of it).
And some use it for Nextcloud and shared some experience:
Lighttpd Nextcloud config - #22 by krs