  In this paper we obtain an explicit formula of Cauchy--Szeg\"{o} kernel for
quaternionic Siegel upper half space, and then based on this, we prove that the
Cauchy--Szeg\"{o} projection on quaternionic Heisenberg group is a
Calder\'on--Zygmund operator via verifying the size and regularity conditions
for the kernel. Next, we also obtain a suitable version of pointwise lower
bound for the kernel, which further implies the characterisations of the
boundedness and compactness of commutator of the Cauchy--Szeg\"{o} operator via
the BMO and VMO spaces on quaternionic Heisenberg group, respectively.
