Can I use persistent connection when using combination of
NGINX + PHP-FPM?
There would be nothing holding you back by doing so (though I have never done it before), but I would highly advise not to use persistent connections in MySQL. It would just give your server way more overhead than what it needs.
Take a look at the below Stack Overflow link for some more info: http://stackoverflow.com/questions/2252612/mysql-persistent-connections
If you want more performance out of your server, take a look into HTTP caching like Varnish. Varnish will have a much heavier impact on your performance than enabling persistent connections.