We study the ground state fidelity, fidelity susceptibility, and quench dynamics of the extended quantum compass model in a transverse field. This model reveals a rich phase diagram which includes several critical surfaces depending on exchange couplings. We present a characterization of quantum phase transitions in terms of the ground state fidelity between two ground states obtained for two different values of external parameters. We also derive scaling relations describing the singular behavior of the fidelity susceptibility in the quantum critical surfaces. Moreover, we study the time evolution of the system after a critical quantum quench using the Loschmidt echo (LE). We find that the revival times of LE are given by , where N is the size of the system and vmax is the maximum of the lower bound group velocity of quasi-particles. Although the fidelity susceptibility shows the same exponent in all critical surfaces, the structure of the revivals after critical quantum quenches displays two different regimes reflecting different equilibration dynamics.