In this paper, we study surfaces of revolution without parabolic points in 3-Euclidean space $mathbb{R}^3$, satisfying the condition ${Delta}^{II}G=f(G+C)$, where ${Delta}^{II}$ is the Laplace operator with respect to the second fundamental form, $f$ is a smooth function on the surface and C is a constant vector. Our main results state that surfaces of revolution without parabolic points in $mathbb{R}^3$ which satisfy the condition ${Delta}^{II}G=fG$, coincide with surfaces of revolution with non-zero constant Gaussian curvature.