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

Characterization of quadric surfaces in terms of coordinate finite type Gauss map