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The first general Zagreb (FGZ) index (also known as the general zeroth-order Randić index) of a graph G can be defined as MγG=∑uv∈EGdGγ−1u+dGγ−1v, where γ is a real number. As MγG is equal to the order and size of G when γ=0 and γ=1, respectively, γ is usually assumed to be different from 0 to 1. In this paper, for every integer γ≥2, the FGZ index Mγ is computed for the generalized F-sums graphs which are obtained by applying the different operations of subdivision and Cartesian product. The obtained results can be considered as the generalizations of the results appeared in (IEEE Access; 7 (2019) 47494–47502) and (IEEE Access 7 (2019) 105479–105488).