Yes, it is possible. You can simply use custom nginx conf to your custom site; though, everything has to be done manually from the terminal.
Take a look at /etc/nginx.conf, and notice that the last line include file from /etc/nginx/conf.d folder.
This mean that you can create custom yoursite.com.conf inside of /etc/nginx/conf.d folder. It may look something like this: https://gist.github.com/chadrien/5394508
This is how VestaCP does it. It has it's own custom nginx-server.conf which is setup on port 8083. It is using php-fpm instead of nginx -> apache -> fcgi for regular sites.