  Topological order is important in many aspects of condensed matter physics,
and has been extended to bosonic systems. In this Letter we report on the
nontrivial topology of the magnon bands in two distinct quantum skyrmion
crystals appearing in zero external magnetic field. This is revealed by nonzero
Chern numbers for some of the bands. As a bosonic analog of the quantum
anomalous Hall effect, we show that topological magnons can appear in skyrmion
crystals without explicitly breaking time-reversal symmetry with an external
magnetic field. By tuning the value of the easy-axis anisotropy at zero
temperature, we find eight quantum topological phase transitions signaled by
discontinuous jumps in certain Chern numbers. We connect these quantum
topological phase transitions to gaps closing and reopening between magnon
bands.
