In this paper, we propose a spectral-Galerkin approximation for the quad-curl eigenvalue problem within spherical geometries. By using the vector spherical harmonics and the Laplace-Beltrami operator, we decouple the quad-curl eigenvalue problem into two classes of independent fourth-order equations: the TE and TM modes. Then, we provide a comprehensive analysis for the TE mode. The TM mode is governed by coupled fourth-order equations subject to a divergence-free condition. We devise two sets of vector basis functions to address the coupled systems in balls and spherical shells, respectively. Additionally, a parameterized method is designed to filter out spurious eigenvalues. Numerical examples are presented to demonstrate the high precision achieved by the proposed method. We also include some graphs to illustrate the localization of eigenfunction. Furthermore, we employ Bessel functions to analyze the quad-curl problem, revealing the intrinsic connection between eigenvalues and zeros of combinations of Bessel functions.