Quantum phonon transport through benzene and alkane chains amide-linked with single wall carbon nanotubes (SWCNTs) is studied within the level of density functional theory. The force constant matrices are obtained from standard quantum chemistry software. The phonon transmission and thermal conductance are from the nonequilibrium Green's function and the mode-matching method. We find that the ballistic thermal conductance is not sensitive to the compression or stretching of the molecular junction. The terminating groups of the SWCNTs at the cutting edges only influence the thermal conductance quantitatively. The conductance of the benzene and alkane chains shows large difference. Analysis of the transmission spectrum shows that (i) the low temperature thermal conductance is mainly contributed by the SWCNT transverse acoustic modes, (ii) the degenerate phonon modes show different transmission probability due to the presence of molecular junction, (iii) the SWCNT twisting mode can hardly be transmitted by the alkane chain. As a result, the ballistic thermal conductance of alkane chains is larger than that of benzene chains below 38 K, while it is smaller at higher temperature.