We study model-based and model-free policy optimization in a class of nonzero-sum stochastic dynamic games called linear quadratic (LQ) deep structured games. In such games, players interact with each other through a set of weighted averages (linear regressions) of the states and actions. In this paper, we focus our attention to homogeneous weights; however, for the special case of infinite population, the obtained results extend to asymptotically vanishing weights wherein the players learn the sequential weighted mean-field equilibrium. Despite the non-convexity of the optimization in policy space and the fact that policy optimization does not generally converge in game setting, we prove that the proposed model-based and model-free policy gradient descent and natural policy gradient descent algorithms globally converge to the sub-game perfect Nash equilibrium. To the best of our knowledge, this is the first result that provides a global convergence proof of policy optimization in a nonzero-sum LQ game. One of the salient features of the proposed algorithms is that their parameter space is independent of the number of players, and when the dimension of state space is significantly larger than that of the action space, they provide a more efficient way of computation compared to those algorithms that plan and learn in the action space. Finally, some simulations are provided to numerically verify the obtained theoretical results.