Symbolic-numeric algorithm for solving the boundary-value problems that describe the model of quantum tunneling of a diatomic molecule through repulsive barriers is described. Two boundary-value problems (BVPs) in Cartesian and polar coordinates are formulated and reduced to 1D BVPs for different systems of coupled second-order differential equations (SCSODEs) that contain potential matrix elements with different asymptotic behavior. A symbolic algorithm implemented in CAS Maple to calculate the required asymptotic behavior of adiabatic basis, the potential matrix elements, and the fundamental solutions of the SCSODEs is elaborated. Comparative analysis of the potential matrix elements calculated in the Cartesian and polar coordinates is presented. Benchmark calculations of quantum tunneling of a diatomic molecule with the nuclei coupled by Morse potential through Gaussian barriers below dissociation threshold are carried out in Cartesian and polar coordinates using the finite element method, and the results are discussed.