Before initiating the repair process, you have to determine the actual spot where the stair tread is creating the sound. You can walk up and down on the stairs and mark the area which squeaks.
Carve a hole on the point where the stair tread rests upon the stringer by using a counter-sunk drill bit but make sure you only drill enough to let the head of the screw sink under the surface of the wood. The counter-sunk drill bit will sink the screw below the surface of the wood and you can will out the hole with wood filler later on.
Take a wood screw and insert it into the hole. Make sure the screw is long enough to penetrate ¾ inches into the stringer when fully tightened.
You can use a screw gun to tighten up the screw but ensure it is set at a low speed and strong torque. This will secure the wood tread tightly and eliminate the irritating squeaky noise.
You have to be a little careful while counter sinking the screws in the tread. You better stop the screw gun when you see the screw head is fully sunk below the surface of the wood and a small indentation is left.
Try walking up and down on the stairs once again to check if the squeak noise has been eliminated or not. If you still can hear the sound, it means the crew has not penetrated to the required depth. Use the screw gun once again to sink the crew further in and tighten up the tread to the stringer.
Use a trowel blade to insert wood filler into the hole. Remove the excess with trowel and let the filler dry for few hours.