You need to configure Nginx work with cgi scripts. Add to munin.conf this parameters:

html_strategy cgi
graph_strategy cgi

It means that html and graphs will be generated by cgi scripts.

Next, confugure your nginx server config:

location ^~ /munin-cgi/munin-cgi-graph/ {
    access_log off;
	fastcgi_split_path_info ^(/munin-cgi/munin-cgi-graph)(.*);
    fastcgi_param PATH_INFO $fastcgi_path_info;
    fastcgi_pass unix:/var/run/munin/fcgi-graph.sock;
    include fastcgi_params;
location /munin/static/ {
    alias /etc/munin/static/;
location /munin/ {
    fastcgi_split_path_info ^(/munin)(.*);
    fastcgi_param PATH_INFO $fastcgi_path_info;
    fastcgi_pass unix:/var/run/munin/fcgi-html.sock;
    include fastcgi_params;

And then start fcgi sockets:

$ spawn-fcgi -s /var/run/munin/fcgi-graph.sock -U www-data -u www-data -g www-data /usr/lib/munin/cgi/munin-cgi-graph
$ spawn-fcgi -s /var/run/munin/fcgi-html.sock -U www-data -u www-data -g munin /usr/lib/munin/cgi/munin-cgi-html

To debug, just add -n parameter. Pleast be aware, in that way your nginx server must work by www-data user. If not, change your configuration or change username and group in the commands.

Here official manual, there is some difference. Instead of cgi-bin I wrote munin-cgi, don’t know why, but on that works for me.