Attach a 1/8-inch countersink auger bit to a screw gun chuck. Drill a hole through the stair tread and into the stringer two inches in from the edge.
Push the bit down enough so that the countersink drills into the tread to the depth of the screw head. Remove the rotating bit and clean out any debris from inside the hole.
Remove the countersink bit and attach a Phillips drive bit to the screw gun. Place the tip of the screw into the hole and affix the drive bit to the screw head.
Place weight on the top of the tread to remove any space between the tread and the stringer. Thread the screw through the tread and into the stair stringer using a screw gun.
Stop threading the screw once the tread is secure and the fastener head sits inside the countersink hole. Place a small amount of wood filler into the hole and allow it to dry. Sand the dry wood filler with 120-grit or higher sandpaper until it matches the surrounding area. Stain or paint the wood filler to match.