  Let $n=2^s p_{1}^{\alpha_{1}}\cdots p_{k}^{\alpha_{k}}$, where $s=0$ or $1$,
$\alpha_i\geq 1$, and the distinct primes $p_i$ satisfy $p_i\equiv 1\pmod{4}$
for all $i=1, \ldots, k$. Let $\mathbb{Z}_n^\ast$ denote the group of units in
the commutative ring $\mathbb{Z}_n$. Recently, we defined a Paley-type graph
$G_n$ of order $n$ as the graph whose vertex set is $\mathbb{Z}_n$ and $xy$ is
an edge if $x-y\equiv a^2\pmod n$ for some $a\in\mathbb{Z}_n^\ast$. The
Paley-type graph $G_n$ resembles the classical Paley graph in a number of ways,
and adds to the list of generalizations of the Paley graph. Computing the
number of cliques of a particular order in a Paley graph or its generalizations
has been of considerable interest. For primes $p\equiv 1\pmod 4$ and
$\alpha\geq 1$, by evaluating certain character sums, we found the number of
cliques of order $3$ in $G_{p^\alpha}$ and expressed the number of cliques of
order $4$ in $G_{p^\alpha}$ in terms of Jacobi sums. In this article we give
combinatorial proofs and find the number of cliques of orders $3$ and $4$ in
$G_n$ for all $n$ for which the graph is defined.
