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The relativistic solutions of the Klein-Gordon equation comprising an interaction of the generalized inversely quadratic Yukawa potential mixed linearly with the hyperbolic Schiöberg molecular potential is achieved employing the idea of parametric Nikiforov-Uvarov and the Greene-Aldrich approximation scheme. The energy spectra and the corresponding normalized wave functions are derived regarding the hypergeometric function in a closed form for arbitrary ℓ -state. Numerical results of the energy eigenvalue are proposed. Moreover, special circumstances of this potential are reviewed, and their energy eigenvalues were assessed. Subsequently, the Tsallis entropy and Rényi entropy both in position and momentum spaces are defined under the desired potential. The impacts of these entropies on the angular momentum quantum number are explored in detail. [ABSTRACT FROM AUTHOR]