  For positive integers $p$ and $q$ let $G:=\textrm{PSO}(p,q)$ be the
projective indefinite special-orthogonal group of signature $(p,q)$. We study
counting problems in the Riemannian symmetric space $X_G$ of $G$ and in the
pseudo-Riemannian hyperbolic space $\mathbb{H}^{p,q-1}$. Let $S\subset X_G$ be
a totally geodesic copy of $X_{\textrm{PSO}(p,q-1)}$. We look at the orbit of
$S$ under the action of a projective Anosov subgroup of $G$. For certain
choices of such a geodesic copy we show that the number of points in this orbit
which are at distance at most $t$ from $S$ is finite and asymptotic to a purely
exponential function as $t$ goes to infinity. We provide an interpretation of
this result in $\mathbb{H}^{p,q-1}$, as the asymptotics of the amount of
space-like geodesic segments of maximum length $t$ in the orbit of a point.
