Let $(X,\tau)$ be any generalized topological space. If $A\subseteq X$ is open and $A\neq \emptyset$, it cannot be be strongly nowhere dense: Let $U=A$, then for every nonempty open set $V\subseteq U$ we have $V \cap A = V\cap U = V \neq \emptyset$.

So $A = \emptyset$ is the only strongly nowhere dense subset of $(X,\tau)$.