There can never be an electric field inside the body of conductor, in other words there will be no electric field just beneath the surface of the conductor. This is because, the entire conductor has to be at the same electric potential. If there were a electric field within the conductor it would mean that there would be a potential difference between two points within the conductor. Let us assume that the charge density on the conductor at a distance r from the point where the perpendicular from the charge q touches the conducting plane is .
Consider an infinitesimally small cylinderical Gaussian Pillbox, of area dA and height dh (considered to be as thin as the thickness of a plane), a this location as shown in the figure below. One flat end surface of this cylinder lies within the thickness of the conductor (it just penetrates the conductor) and other flat end is just above the surface of the conductor.