An efficient topology optimization method for fluid-structure problems was developed in an effort to determine the optimum flow channel route in a fuel cell bipolar plate from first principles. This study describes the derivation and solution of new mathematical equations for topology optimization combining a density-based algorithm, the interpolation method of moving asymptotes (MMA), and the incompressible Navier-Stokes equation with a term representing the chemical reaction between hydrogen and the catalyst. The present method is based on the finite element method with a newly developed reaction rate equation. In this model, a topology variable of 0 represents viscous flow, whereas a value of 1 indicates porous flow. The flow velocity and pressure were obtained from the Navier-Stokes equation and constraints and element matrices for sensitivity analyses during the optimization. MMA was utilized to calculate the optimum flow routes in the design domain. The influence of the key design parameter q and the pressure drop on the optimum topology were also investigated. The channel topology became smoother with decreasing q, and the number of channels increased with increasing pressure drop.