My Fax machine which is always off, is reportedly making 911 calls and I'm being visited by the local police nightly.
Is there even the slightest possibility that your fax machine is under attack? These are dangerous times.
More seriously, without further information it's impossible to give a definitive answer. For example, is your fax machine a standalone device, is it part of your computer, is it contained in some sort of telephone device, etc.?
Also, these machines typically will continue to try to send a fax if they do not at first succeed. Is there any possibility that something like this is going on?
Is there a chance that at some point in the past the machine actually did try to call a number that contains the "9-1-1" sequence. This happened to me on more than one occasion due to a dialing error.
If you give us more specifics, then perhaps someone here will be able to address your issue.