[en] The quantum criticality of the two-lead two-channel pseudogap Anderson impurity model is studied. Based on the non-crossing approximation (NCA) and numerical renormalization group (NRG) approaches, we calculate both the linear and nonlinear conductance of the model at finite temperatures with a voltage bias and a power-law vanishing conduction electron density of states, ${\mathrm{\rho <!--\; ??\; -->}}_{\text{c}}(\mathrm{\omega <!--\; ??\; -->})\propto <!--\; ???\; -->|\mathrm{\omega <!--\; ??\; -->}-<!--\; ???\; -->{\mathrm{\mu <!--\; ??\; -->}}_{\text{F}}{|}^r$ (0  <  r  <  1) near the Fermi energy ${\mathrm{\mu <!--\; ??\; -->}}_{\text{F}}$. At a fixed lead-impurity hybridization, a quantum phase transition from the two-channel Kondo (2CK) to the local moment (LM) phase is observed with increasing r from r  =  0 to $r=r_{\text{c}}<1$. Surprisingly, in the 2CK phase, different power-law scalings from the well-known $\sqrt{T}$ or $\sqrt{V}$ form is found. Moreover, novel power-law scalings in conductances at the 2CK-LM quantum critical point are identified. Clear distinctions are found on the critical exponents between linear and non-linear conductance at criticality. The implications of these two distinct quantum critical properties for the non-equilibrium quantum criticality in general are discussed. (paper)