We report ab initio calculations of the indirect nuclear spin-spin coupling constants of PbH₄ using a basis set which was specially optimized for correlated calculations of spin-spin coupling constants. All nonrelativistic contributions and the most important part of the spin-orbit correction were evaluated at the level of the random phase approximation. Electron correlation corrections to the coupling constants were calculated with the multiconfigurational linear-response method using extended complete and restricted active space wavefunctions as well as with the second-order polarization propagator approximation and the second-order polarization propagator approximation with coupled-cluster singles and doubles amplitudes. The effects of nuclear motion were investigated by calculating the coupling constants as a function of the totally symmetric stretching coordinate. We find that the Fermi contact term dominates the Pb-H coupling, whereas for the H-H coupling it is not more important than the orbital paramagnetic and diamagnetic contributions. Correlation affects mainly the Fermi contact term. Its contribution to the one-bond coupling constant is reduced by correlation, independent of the method used; however, the different correlated methods give ambiguous results for the Fermi contact contribution to the H-H couplings. The dependence of both coupling constants on the Pb-H bond length is dominated by the change in the Fermi contact term. The geometry dependence is, however, overestimated in the random phase approximation.