Quincy House Apartments For Rent . We found 1200 apartments for rent in quincy, ma. This apartment building is located at 229 quarry st., quincy, ma 02169. 934 E Squantum St, Quincy, MA 3 Bed, 2 Bath SingleFamily Home 8 from www.trulia.com Towne house apartments is located in quincy, massachusetts in the 02169 zip code. Compare rentals, see map views and save your favorite houses. Compare rentals, see map views and save your favorite apartments.