  It is known that the number of directions formed by a Cartesian product $A
\times B \subset AG(2,p)$ is at least $|A||B| - \min\{|A|,|B|\} + 2$, provided
$p$ is prime and $|A||B|<p$. This implies the best known upper bound on the
clique number of the Paley graph over $\mathbb{F}_p$. In this paper, we extend
this result to $AG(2,q)$, where $q$ is a prime power. We also give improved
upper bounds on the clique number of generalized Paley graphs over
$\mathbb{F}_q$. In particular, for a cubic Paley graph, we improve the trivial
upper bound $\sqrt{q}$ to $0.769\sqrt{q}+1$. In general, as an application of
our key result on the number of directions, for any positive function $h$ such
that $h(x)=o(x)$ as $x \to \infty$, we improve the trivial upper bound
$\sqrt{q}$ to $\sqrt{q}-h(p)$ for almost all non-squares $q$.
