What are the requirements for FRET?
A low quantum yield would indicate that there are efficient intra-molecular non-radiative decay pathways, likely internal conversion or spin-orbit coupling. These decay pathways may be faster that FRET, rendering FRET essentially a non-competetive process.

A small extinction coefficient would indicate a small transition dipole moment. If the transition dipole moment is small it is difficult for the FRET partner to transfer energy since it is a weak transition resulting in a small coupling  between the two chromophores.