  Let $q$ be a prime power such that $q\equiv 1\pmod{4}$. The Paley graph of
order $q$ is the graph with vertex set as the finite field $\mathbb{F}_q$ and
edges defined as, $ab$ is an edge if and only if $a-b$ is a non-zero square in
$\mathbb{F}_q$. We attempt to construct a similar graph of order $n$, where $n$
is a positive integer. For suitable $n$, we construct the graph where the
vertex set is the finite commutative ring $\mathbb{Z}_n$ and edges defined as,
$ab$ is an edge if and only if $a-b\equiv x^2\pmod{n}$ for some unit $x$ of
$\mathbb{Z}_n$. We look at some properties of this graph. For primes $p\equiv
1\pmod{4}$, Evans, Pulham and Sheehan computed the number of complete subgraphs
of order $4$ in the Paley graph of order $p$. Very recently, Dawsey and
McCarthy find the number of complete subgraphs of order 4 in the generalized
Paley graph of order $q$. In this article, we find the number of complete
subgraphs of order 3 and 4 in our graph defined over $\mathbb{Z}_n$.
