In this work, we investigate the construction of a new discontinuous Galerkin discrete formulation to approximate the solution of Serre-Green-Naghdi (SGN) equations in the one-dimensional horizontal framework. Such equations describe the time evolution of shallow water free surface flows in the fully nonlinear and weakly dispersive asymptotic approximation regime. A new non conforming discrete formulation belonging to the family of Symmetric Interior Penalty discontinuous Galerkin methods (SIP-DG) is introduced to accurately approximate the solutions of the second order elliptic operator occurring in the SGN equations. We show that the corresponding discrete bilinear form enjoys some consistency and coercivity properties, thus ensuring that the corresponding discrete problem is well-posed. The resulting global discrete formulation is then validated through an extended set of benchmarks, including convergence studies and comparisons with data taken from experiments.