To answer your subject line, no. A .mozilla folder in home directories does not mean any mozilla process is running, only that a mozilla application is installed and set up.
So are you sitting at this machine, or is this a remote server? Assuming the latter, since you would be able to easily see if a GUI is running on a local machine...
Running the command 'runlevel' might help. If the result is '5' you are probably running an X login, '3' it is probably in multi-user text mode, though a user can still run an X session. These numbers may be different for different distros though.