Apartments For Rent In Adams County Co . The average voucher holder contributes $300 towards rent in adams county. Adams county co apartments for rent. Denizen Denver, CO Apartment Finder from www.apartmentfinder.com 1 in the county of denver and state of c. On average, section 8 housing choice vouchers pay adams county landlords $1,000 per month towards rent. Use our detailed filters to find the perfect place, then get in touch with the property manager.