It sounds like there's an emergency generator on site that, when there's a fire alarm, gets turned on. It also sounds like when the fire alarm system is silenced, but still in alarm, the generator turns off.
If that's the case, that would mean that the fire-alarm-system-to-generator interface is connected to a horn circuit (Notification Appliance Circuit or NAC) and not directly to an alarm output on the fire alarm panel. This connection would turn on the generator whenever the horns turn on and turn off the generator when the horns turn off.
I take it that the generator is supposed to remain of as long as the fire alarm system is in alarm.
To do that, generator control wires have to be run all the way to the fire alarm control panel and be connected to a fire alarm output relay. Connecting the generator control circuit is a closer-to-correct method of connecting to the generator than using the NAC circuit, anyway.
Before running the wires, talk to the fire alarm service company to find out if there is even a place to connect these wires in the current fire alarm panel.
As a alternative to running wires, maybe a timer could be added to the interface to keep the generator running for some time after a fire alarm (30 minutes?).
Of course, there also may be a timer already in the generator for this.