Recently, quantum logic gates and quantum algorithms have been experimentally realized within some magnetic molecules including single-molecule magnets. For such applications, either molecular electronic levels or nuclear levels can be used for quantum bits and they can be read, operated, or written using magnetic or electric fields. In particular, lanthanide-based single-molecule magnets are promising for such applications due to strong spin-orbit coupling, molecular geometry compatible to devices, and tunability of coupling between electron and nuclear spins via external means. Motivated by the recent experimental work, we investigate the interaction between the electronic magnetic moment and the lanthanide-element nuclear spin in trivalent and divalent terbium-based single-molecule magnets with and without an electric field, using multiconfigurational ab-initio methods (quantum chemistry methods) including spin-orbit coupling. We show that trivalent and divalent terbium-based molecules have different types of dominant contributions to the magnetic hyperfine interaction, and that the principal axes of the nuclear quadrupole interaction are not necessarily parallel to those of the magnetic axes. We further examine how the magnetic hyperfine and nuclear quadrupole interactions are influenced by the electric field in the terbium-based single-molecule magnets, as a function of the magnitude and direction of the electric field.