In this paper, we investigate a distributed Nash equilibrium computation problem for a time-varying multi-agent network consisting of two subnetworks, where the two subnetworks share the same objective function. We first propose a subgradient-based distributed algorithm with heterogeneous stepsizes to compute a Nash equilibrium of a zero-sum game. We then prove that the proposed algorithm can achieve a Nash equilibrium under uniformly jointly strongly connected (UJSC) weight-balanced digraphs with homogenous stepsizes. Moreover, we demonstrate that for weighted-unbalanced graphs a Nash equilibrium may not be achieved with homogenous stepsizes unless certain conditions on the objective function hold. We show that there always exist heterogeneous stepsizes for the proposed algorithm to guarantee that a Nash equilibrium can be achieved for UJSC digraphs. Finally, in two standard weight-unbalanced cases, we verify the convergence to a Nash equilibrium by adaptively updating the stepsizes along with the arc weights in the proposed algorithm.