You're never going to have a FACP system activate a sprinkler system, it's always going to be the exact reverse scenario.
Usually there's going to be a flow switch and possibly a pressure switch. There is usually some sort of requirement for a bell or similar outside, independent of any supervising system, to alert whoever that the sprinklers are flowing.
In your case, I'd probably go with the fact the residence is probably too far from the closest fire house, so there is a local mandate for a sprinkler installation. We have a few locales that require the same around here.
The HAI system will not be an initiating device for sprinkler activation, only supervising and reporting items such as flow and/or low air (if the designed system is pressurized in that manner). You would need to discuss what the sprinkler guys are providing, however their system is generally going to be local only and most fire alarm supervisiory devices have multiple form C contacts to facilitate multiple system supervision methods.