  Let $GP(q,d)$ be the $d$-Paley graph defined on the finite field
$\mathbb{F}_q$. It is notoriously difficult to improve the trivial upper bound
$\sqrt{q}$ on the clique number of $GP(q,d)$. In this paper, we investigate the
connection between Gauss sums over a finite field and maximum cliques of their
corresponding generalized Paley graphs. In particular, we show that the trivial
upper bound on the clique number of $GP(q,d)$ attains if and only $d \mid
(\sqrt{q}+1)$, which strengthens the previous related results by
Broere-D\"oman-Ridley and Schneider-Silva, as well as improves the trivial
upper bound on the clique number of $GP(q,d)$ when $d \nmid (\sqrt{q}+1)$.
