Measure from the inside corner of the wall to the outside corner with a tape measure. Cut one end of the quarter round molding at a 45-degree inside angle with a miter saw. Transfer the wall measurement to the molding and cut the opposite end with an outside corner with the miter saw.
Clean the brick wall near the floor with a stiff brush to remove any loose pieces of brick and mortar. Wipe the area with a damp cloth.
Place the quarter round molding on the floor and against the brick. Drive a brad nail into one end, and into the floor, with a small hammer until the nail head is at the surface of the quarter round.
Go near the center of the length of the quarter round and press it against the brick wall. Tack another brad nail into the quarter round and the floor. Repeat at the opposite end of the quarter round.
Place brad nails through the quarter round and into the floor every 6 inches. Countersink each of the nail heads with a nail set.
Apply a bead of silicone caulk between the brick wall and the quarter round with a caulk gun. Repeat the installation and finishing process for the remaining quarter round. Do not add caulk, however, if the wall is part of a fireplace.