  Let $\Omega$ be a bounded domain (with smooth boundary) on the hyperbolic
plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the
$(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$. In this
paper, by using a priori estimates, we can establish Pogorelov type estimates
of $k$-convex solutions to a class of Hessian quotient equations defined over
$\Omega\subset\mathscr{H}^{n}(1)$ and with the vanishing Dirichlet boundary
condition.
