Let G=(V,E) be a simple connected graph. The sets of vertices and edges of G are denoted by V=V(G) and E=E(G), respectively. A topological index of a graph is a number related to a graph which is invariant under graph automorphisms. In chemical graph theory, we have many invariant polynomials and topological indices for a molecular graph. In 1972, Gutman and Trinajstić introduced the First and Second Zagreb topological indices of molecular graphs. The First and Second Zagreb indices are equal to M1(G)= vÎV(G) Σ dv 2 and M2(G)= uvÎE(G) Σ (du×dv), respectively. These topological indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. In this paper, we focus on the structure of V-Phenylenic Nanotubes and Nanotori and compute the Generalized Zagreb index of these Nanostructures.