Orbital degrees of freedom in condensed matter could play important roles in forming a variety of exotic electronic states by interacting with conduction electrons. In 4f-electron systems, because of strong intra-atomic spin-orbit coupling, an orbitally degenerate state inherently carries quadrupolar degrees of freedom. The present work has focused on a purely quadrupole-active system PrIr2Zn20 showing superconductivity in the presence of an antiferroquadrupole order at T-Q = 0.11 K. We observed non-Fermi-liquid (NFL) behaviors emerging in the electrical resistivity rho and the 4f contribution to the specific heat, C-4f, in the paramagnetic state at T > T-Q. Moreover, in magnetic fields B <= 6 T, all data sets of rho(T) and C-4f (T) are well scaled with characteristic temperatures T-0's. This observation of the NFL state in the nonmagnetic quadrupole-active system has an origin intrinsically different from that observed in the vicinity of the conventional quantum critical point. It implies possible formation of a quadrupole Kondo lattice resulting from hybridization between the quadrupoles and the conduction electrons with an energy scale of k(B)T(0). At T <= 0.13 K, rho(T) and C-4f (T) exhibit anomalies as B approaches 5 T. This is the manifestation of a field-induced crossover toward a Fermi-liquid ground state in the quadrupole Kondo lattice.