Look carefully at that page; it explains several things you can do. One of those is *not* to run an actual X server -- it's not going to be a convenient solution, and doesn't do much for your site security.
As far as finding out if an X server is running, type
ps -ef | grep X
and look for a process just named "X". But even if one
is running, unless it's running as the same user that the server is running as, chances are it's set up with security options to prevent the server from connecting, anyway.
So your best option is to use the "-Djava.awt.headless=true
Java command-line switch when starting your Java server.