You have an analog phone line that has bad wiring. Maybe a rusty outlet, or a cable that's partially damaged by water, or other physical means.
Then what happens is that at the point where the cable or outlet is damaged/rusty, you have occasional short-circuits on the line. And I don't care how unbelievable it may sound, this happened to me already: the cable short-circuits at random times, on and off, so the line goes off-hook and then on hook multiple times in a minute or even in a second.
There is some mechanism that when this happens, a 911 call is made. Maybe the short-circuits act as dial pulses, and they are translated into digits 911. Maybe you have short-circuits on this line all the time, and occasionally they go through as a 911 call.
But when a 911 call gets originated, your dedicated 911 trunks gets seized, and the emergency response center gets the call. BUT, before they answer the call, the originating station with its faulty wiring goes back off-hook (hangs up), so there is not a completed/answered call, so SMDR doesn't report it. But the Emergency responders do get the caller ID of your call even if your "caller" hangs up before they answer, so they'll call you to find out what's happening.
To resolve this, go to your system, and hopefully, on the line-cards, you have LED-s corresponding to stations, and the LEDs are:
off: if the station is idle
on: when the station is in-use
blinking: when the port is not programmed
Look for and LED that's mostly on, or always on, or blinking erratically. This might take a while to find, but this is your way of finding it. The corresponding phone is at fault. You should disconnect the cross-connect wire from that port so no more calls are made, and then either pull in new cable, or find out what's the fault with the existing one. Probably a rusty outlet.
Here is what happened on my site:
Example 1: A building underwent some construction. Some electrician gutted the walls, and left a phone cable, that had a live analog station-port still connected to it, left it laying on the ground, in a puddle of water. This line called 911 twice within a week.
Example 2: I had an intercom phone at an entrance gate. There was "indoor" cat 5 going to that phone, under-ground, in conduits, but in one of the pull-boxes in the grass, there was a bee-hive. They put honey all over the cable and somehow that honey (or the bees?) got through the jacket of the cable and created the intermittent short-circuit on that line. So my gate-intercom called 911.