Local f-electron spin dynamics in the non-Fermi-liquid heavy-fermion alloys UCu5-xPdx, x = 1.0 and 1.5, have been studied using muon spin-lattice relaxation. The sample-averaged asymmetry function G(t) indicates strongly inhomogeneous spin fluctuations and exhibits the scaling G(t,H) = G(t/H(gamma)) expected from glassy dynamics. At 0.05 K gamma(x = 1.0) = 0.35+/-0.1, but gamma(x = 1.5) = 0.7+/-0.1. This is in contrast to inelastic neutron scattering results, which yield gamma = 0.33 for both concentrations. There is no sign of static magnetism approximately greater than 10(-3)(B)/U ion in either material above 0.05 K. Our results strongly suggest that both alloys are quantum spin glasses.