I think your question is more a question of how to administer your server than it is a question about Perl.

The basic steps to do so would be:

1. Install a headless X server, like Xvfb through whatever package manager your system has.
2. Install Firefox
3. Run Firefox on the display provided by the headless X server.
4. Check that Firefox is behaving properly by connecting to your X server through (for example) vnc

I have not done this myself, but others report great success in doing so.