Summary: In this article, we introduce an important class of surfaces, namely, quadrics in the Euclidean 3-space \(\mathbb{E}^3\). We prove that spheres, planes, and circular cylinders are the only quadric surfaces whose Gauss map \(n\) satisfies a relation of the form \(\Delta^In=Mn\), where \(M\) is a square matrix of order 3 and \(\Delta^I\) is the Laplace-Beltrami operator corresponding to the first fundamental form \(I\) of the surface.