Fire Alarm Systems as covered in NFPA 72 require a secondary power supply, not an "emergency source", nor is it classed as a "legally required standby source". Fire alarm secondary power is usually accomplished via batteries. Check out Chapter 4 of NFPA 72.
Only those loads required to be emergency loads are allowed to be connected to emergency sources. (NEC 700.9(B))
So, no, you cannot use an existing emergency source for the fire alarm system.
However, there is not a prohibition for loads cannected to the "legally required standby source".
But, NFPA 72, in section 126.96.36.199.3.1 requires engine-driven generators to comply with NFPA 110, for a Type 10, Class 24, Level 1 System. Type 10 means the generator starts and transfers within 10 seconds. Class 24 means there is enough fuel to operate at full load for 24 hours. Level 1 systems shall be installed when failure of the equipment to perform could result in loss of human life or serious injuries.
And, NFPA 72 section 188.8.131.52.1 requires a four hour battery system if a prime mover provides the power for continuity of service.
So, the bottom line: Use batteries, forget the generator.