All accounts, claims and demands of whatever nature existing against any county for opening, constructing, maintaining and operating any public highway, road, bridge or ferry shall be presented to the governing body of the county duly attested, and, if approved by said body, the county supervisor shall draw his warrant upon the county treasurer under the seal of the governing body of the county for the amount of any such claim. Such warrant shall be countersigned by the secretary of the governing body and shall be paid by the county treasurer out of the county road fund.