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We study the transport properties of a Josephson junction consisting of two identical s-wave superconductors separated by an even-layer MnBi_{2}Te_{4} (MBT). Using recursive Green's function method, we calculate the supercurrent in the presence of a perpendicular magnetic field and find that its quantum interference exhibits distinct patterns when the MBT is in different magnetic states. In the antiferromagnetic state, the MBT is an axion insulator supporting an extended “hinge” supercurrent, which leads to a sinusoidal interference pattern decaying with the field strength. In the ferromagnetic state, the MBT is a Chern insulator and the unbalanced chiral supercurrents on opposite edges give rise to a highly asymmetric interference pattern. If the MBT turns into a metal as the Fermi level is tuned into the conduction band, the interference exhibits a Fraunhofer pattern due to the uniformly distributed bulk supercurrent. Our work unravels a strong indicator to identify different phases in the MBT and can be verified directly by experiments.