Sensitive signal detection and processing in classical world, especially in quantum regime, require nonreciprocal manipulation. In this paper we show how to achieve nonreciprocal quantum synchronization for two magnon modes in a two-sublattice antiferromagnet with strong isolation.The antiferromagnet is trapped in a cavity with two posts so that the two magnon modes not only couple to each other through a parametric-type interaction, but also interact with a same cavity respectively in a beam splitter-type and parametric-type ways. Under the condition of systems stability, we show that nonreciprocal quantum synchronization between two magnon modes is mainly dependent on resonance frequency of the cavity caused by direction of input currents. In addition,quantum synchronization is enhanced by the increase of interaction strength between two Bogoliubov modes and cavity mode. Moreover, numerical simulation results with parameters commonly used in current experiments show that the present scheme may be feasible.
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