We study the transmission problem for the Navier-Stokes and for the Darcy-Forchheimer-Brinkman systems with smooth coefficients in two complementary Lipschitz domains in a closed Riemannian manifold. First, we use a boundary integral method to show the well-posedness of the transmission problem for the Stokes system in \(L^2\) -based Sobolev spaces on manifolds of dimension greater or equal 2 and in their Lipschitz domains. Then, using this well-posedness result and a fixed point theorem, we obtain an existence and uniqueness result for the transmission problem for the nonlinear Navier-Stokes and Darcy-Forchheimer-Brinkman systems with small \(L^2\) data for manifolds of dimension 2 or 3.