  Let $k \geq 2$ be an integer. Let $q$ be a prime power such that $q \equiv 1
\pmod {k}$ if $q$ is even, or, $q \equiv 1 \pmod {2k}$ if $q$ is odd. The
generalized Paley graph of order $q$, $G_k(q)$, is the graph with vertex set
$\mathbb{F}_q$ where $ab$ is an edge if and only if ${a-b}$ is a $k$-th power
residue. We provide a formula, in terms of finite field hypergeometric
functions, for the number of complete subgraphs of order four contained in
$G_k(q)$, $\mathcal{K}_4(G_k(q))$, which holds for all $k$. This generalizes
the results of Evans, Pulham and Sheehan on the original ($k$=2) Paley graph.
We also provide a formula, in terms of Jacobi sums, for the number of complete
subgraphs of order three contained in $G_k(q)$, $\mathcal{K}_3(G_k(q))$. In
both cases we give explicit determinations of these formulae for small $k$. We
show that zero values of $\mathcal{K}_4(G_k(q))$ (resp.
$\mathcal{K}_3(G_k(q))$) yield lower bounds for the multicolor diagonal Ramsey
numbers $R_k(4)=R(4,4,\cdots,4)$ (resp. $R_k(3)$). We state explicitly these
lower bounds for small $k$ and compare to known bounds. We also examine the
relationship between both $\mathcal{K}_4(G_k(q))$ and $\mathcal{K}_3(G_k(q))$,
when $q$ is prime, and Fourier coefficients of modular forms.
