  For a prime $p\equiv 3\pmod{4}$ and a positive integer $t$, let $q=p^{2t}$.
Let $g$ be a primitive element of the finite field $\mathbb{F}_q$. The Peisert
graph $P^\ast(q)$ is defined as the graph with vertex set $\mathbb{F}_q$ where
$ab$ is an edge if and only if $a-b\in\langle g^4\rangle \cup g\langle
g^4\rangle$. We provide a formula, in terms of finite field hypergeometric
functions, for the number of complete subgraphs of order four contained in
$P^\ast(q)$. We also give a new proof for the number of complete subgraphs of
order three contained in $P^\ast(q)$ by evaluating certain character sums. The
computations for the number of complete subgraphs of order four are quite
tedious, so we further give an asymptotic result for the number of complete
subgraphs of any order $m$ in Peisert graphs.
