  The cross-correlation study of the unresolved $\gamma$-ray background (UGRB)
with galaxy clusters has a potential to reveal the nature of the UGRB. In this
paper, we perform a cross-correlation analysis between $\gamma$-ray data by the
Fermi Large Area Telescope (Fermi-LAT) and a galaxy cluster catalogue from the
Subaru Hyper Suprime-Cam (HSC) survey. The Subaru HSC cluster catalogue
provides a wide and homogeneous large-scale structure distribution out to the
high redshift at $z=1.1$, which has not been accessible in previous
cross-correlation studies. We conduct the cross-correlation analysis not only
for clusters in the all redshift range ($0.1 < z < 1.1$) of the survey, but
also for subsamples of clusters divided into redshift bins, the low redshift
bin ($0.1 < z < 0.6$) and the high redshift bin ($0.6 < z < 1.1$), to utilize
the wide redshift coverage of the cluster catalogue. We find the evidence of
the cross-correlation signals with the significance of 2.0-2.3$\sigma$ for all
redshift and low-redshift cluster samples. On the other hand, for high-redshift
clusters, we find the signal with weaker significance level (1.6-1.9$\sigma$).
We also compare the observed cross-correlation functions with predictions of a
theoretical model in which the UGRB originates from $\gamma$-ray emitters such
as blazars, star-forming galaxies and radio galaxies. We find that the detected
signal is consistent with the model prediction.
