We propose a new operator splitting algorithm for distributed Nash equilibrium seeking under stochastic uncertainty, featuring relaxation and inertial effects. The proposed algorithm is derived from a forward-backward-forward scheme for solving structured monotone inclusion problems with Lipschitz continuous and monotone pseudogradient operator. To the best of our knowledge, this is the first distributed generalized Nash equilibrium seeking algorithm featuring acceleration techniques in stochastic Nash equilibrium problems without assuming cocoercivity. Numerical examples illustrate the effect of inertia and relaxation on the performance of our proposed algorithm.