The equilibrium Rayleigh-Brillouin spectrum for a Maxwell fluid in which heat conduction is governed by the classical Fourier law is computed. This is done by using fluctuating hydrodynamics and by the alternative procedure used by Mountain. It is found that both calculations render the same results only if the fluctuation-dissipation relations in the former are the ones prescribed by Landau and Lifshitz for the case where a dispersive dissipation coefficient is present. This result brings up the question of whether extended irreversible thermodynamics is the proper underlying nonequilibrium thermodynamic framework for the Maxwell fluid. It also suggests that at least the version of fluctuating hydrodynamics based on extended irreversible thermodynamics as introduced by Jou and Casas-Vázquez many years ago  must be revised, since it leads to a different form for the fluctuation-dissipation relations.