Kind of. The emergency egress lighting is required to turn on when the normal lighting in a space is lost, not just in a utility outage. With unit equipment or luminaire mounted batteries, it is trivial to use the battery charging circuit as the sensing circuit so that is how the equipment is built. When there is no voltage on the battery charging circuit, the equipment knows normal power is lost and turns on.
Since the inverter output feeds multiple areas with multiple normal circuits feeding the normal lighting, this same approach won't work. If you sense only the input circuit, you would know if there was a utility outage or if the input circuit lost power, but you wouldn't know if one of the normal lighting circuits had lost power which could leave an area in darkness. The UL 924 or 1008 relays move the power loss sensing to the individual normal lighting circuits to allow the emergency egress lights to turn on if that normal circuit loses power. Your inverter typically doesn't get fed from the same circuits feeding the normal lighting because there are typically multiple circuits. A relay gets provided for each normal lighting circuit, and usually each control zone if you want the emergency egress lighting to follow the normal lighting switching, that will have a voltage sensing circuit to force the emergency egress lighting on upon loss of the normal lighting circuit in the area.
The output of an inverter gets wired similarly to emergency egress lighting fed from a generator.