WWW::Mechanize::Firefox and Firebug are completely different. As you don't show any of the expected output you seem to see in Firebug, there is little to do to help you. Please take the time to work on your question skills.

WWW::Mechanize::Firefox does not know abohut Firebug, so you will have to do the necessary legwork yourself.

Update: Upon further reading, you seem to be very mistaken about the nature of the $res object. Using Data::Dumper will not work on it, or rather, will not return much information on it, as it is synthesized from the Javascript results. You will have to learn about Firefox and Javascript to make much sense of the results.