Measure the length of the stair landing edge with a tape measure. Cut a piece of quarter-round molding to the measurement with a coping saw. If you have a landing edge with several sides, measure and cut all pieces for the molding. Use 45-degree cuts where two pieces of molding will join at the ends.
Drill pilot holes through the center face of the quarter-round with a 5/16-inch drill bit. Place a hole on each end and in the center. Space two more holes evenly between the center and end holes. This helps prevent splitting the quarter-round molding when nailing it against the stair landing wall.
Place a straightedge flat on the floating floor and against the stair landing wall. Position the the quarter-round against the stair landing wall, resting it on the straightedge. The straightedge acts as a temporary spacer between the floor and the quarter-round.
Insert finishing brads into the pilot holes on the quarter-round and drive them into the stair landing wall with a hammer. Once the quarter-round is secure, slide the straightedge out from between the floor and bottom of the quarter-round.