In this study, we propose a verified Nyström method for second-kind boundary integral equations on smooth closed curves arising from two-dimensional boundary value problems. The approach establishes a validated-numerics framework for second-kind Fredholm integral equations in which well-posedness and error bounds are obtained via a Neumann-series argument (Banach's fixed point theorem), and all quantities required in the bound are evaluated using interval arithmetic. We apply the proposed method to the two-dimensional Laplace equation and demonstrate how it provides a posteriori error bound for the computed solution.