Does our Association’s earthquake coverage have to be with the same agent or broker that writes the underlying fire insurance?

No. Actually, you’ll find that the coverage is often split between two different brokers. There are can be advantages to having two different brokers and adjusting teams involved in your loss. A fire that could follow an earthquake event is a very real concern. The agent/broker writing the Association’s Master fire policy would respond only to those portions of the premises impacted by fire, while the earthquake insurance broker (and their adjusting team) would be able to just concentrate on the resulting earthquake damage.