The Enskog kinetic theory of multicomponent granular suspensions employed previously [Gómez González, Khalil, and Garzó, Phys. Rev. E 101, 012904 (2020)2470-004510.1103/PhysRevE.101.012904] is considered further to determine the four transport coefficients associated with the heat flux. These transport coefficients are obtained by solving the Enskog equation by means of the application of the Chapman-Enskog method around the local version of the homogeneous state. Explicit forms of the heat flux transport coefficients are provided in steady-state conditions by considering the so-called second Sonine approximation to the distribution function of each species. Their quantitative variation on the control parameters of the mixture (masses and diameters, coefficients of restitution, concentration, volume fraction, and the background temperature) is demonstrated and the results show that in general the dependence of the heat flux transport coefficients on inelasticity is clearly different from that found in the absence of the gas phase (dry granular mixtures). As an application of the general results, the stability of the homogeneous steady state is analyzed by solving the linearized Navier-Stokes hydrodynamic equations. The linear stability analysis (which holds for wavelengths long compared with the mean free path) shows that the transversal and longitudinal modes are always stable with respect to long-enough wavelength excitations. This conclusion agrees with previous results derived for monocomponent and (dilute) bidisperse granular suspensions but contrasts with the instabilities found in previous works in dry (no gas phase) granular mixtures.