This paper focuses on exploring the convergence properties of a generic player's trajectory and empirical measures in an N-player Linear-Quadratic-Gaussian Nash game, where Brownian motion serves as the common noise. The study establishes three distinct convergence rates concerning the representative player and empirical measure. To investigate the convergence, the methodology relies on a specific decomposition of the equilibrium path in the N-player game and utilizes the associated Mean Field Game framework.