On strongly regular extensions of generalized quadrangles

The parameters $s$, $t$ of a generalized quadrangle are said to be classical if $s=p^\alpha$ and $t=p^\beta$ for some prime number $p$ and nonnegative integers $\alpha$ and $\beta$. A escription is obtained for the possible parameters of strongly regular graphs in which the neighborhoods of the vertices are generalized quadrangles with quasiclassical parameters.