A scheme to generate phonon blockade in a superconducting Josephson junction circuit system is proposed, where the photon, phonon, and artificial two‐level atom are bound into a tripartite interaction. The phonon blockade in the proposal mediated by the two‐level atom differs from the phonon blockade resulting from the single phonon resonance and the destructive interference between different paths, which can be generated within a weak coupling region and a large range of cavity detuning, even with a large optical dissipation rate. In addition, under certain condition, the unconventional photon blockade can be achieved but the phonon blockade is still maintained. Based on the phonon blockade, it is shown that the controlled‐Z gate (CZ) can be implemented with the tripartite interaction.