A Brownian dynamics simulation technique is presented where a Fourier-based N log N approach is used to calculate hydrodynamic interactions in confined flowing polymer systems between two parallel walls. A self-consistent coarse-grained Langevin description of the polymer dynamics is adopted in which the polymer beads are treated as point forces. Hydrodynamic interactions are therefore included in the diffusion tensor through a Green's function formalism. The calculation of Green's function is based on a generalization of a method developed for sedimenting particles by Mucha et al. [J. Fluid Mech. 501, 71 (2004)]. A Fourier series representation of the Stokeslet that satisfies no-slip boundary conditions at the walls is adopted; this representation is arranged in such a way that the total O(N2) contribution of bead-bead interactions is calculated in an O(N log N) algorithm. Brownian terms are calculated using the Chebyshev polynomial approximation proposed by Fixman [Macromolecules 19, 1195 (1986); 19, 1204 (1986)] for the square root of the diffusion tensor. The proposed Brownian dynamics simulation methodology scales as O(N1.25 log N). Results for infinitely dilute systems of dumbbells are presented to verify past predictions and to examine the performance and numerical consistency of the proposed method.