Yes, normally the car owner's insurance (so your girlfriend's policy) would cover the damage you caused to the city's fire hydrant in her car as long as there are no exclusions in her policy that would keep the coverages from extending to you as driver.

The claim by the city should go through her property damage liability (PD) coverage which covers you if your car damages someone else's property. Usually it is their car, but it could be a fence, a house or any other property damaged in an accident - such as the fire hydrant.

In general insurance follows a car first and driver second thus it would be your girlfriend's policy that would cover the damage to the hydrant and then your insurance would be secondary if the cost of repairs exceeded her PD liability limits.

Your girlfriend should report the accident to her insurance agent and make sure there are not any exclusions on her policy that would keep the claim from being accepted since you were driving her vehicle.

This seems to happen in San Francisco a lot.