  Suppose we have a finite thick generalised quadrangle whose automorphism
group $G$ acts primitively on both the set of points and the set of lines. Then
$G$ must be almost simple. In this paper, we show that $\operatorname{soc}(G)$
cannot be isomorphic to $\operatorname{Sz}(2^{2m+1})$ or
$\operatorname{Ree}(3^{2m+1})$ where $m$ is a positive integer.
