  In this paper, we consider the evolution of spacelike graphic hypersurfaces
defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center
at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space
$\mathbb{R}^{n+1}_{1}$ along the inverse Gauss curvature flow (i.e., the
evolving speed equals the $(-1/n)$-th power of the Gaussian curvature) with the
vanishing Neumann boundary condition, and prove that this flow exists for all
the time. Moreover, we can show that, after suitable rescaling, the evolving
spacelike graphic hypersurfaces converge smoothly to a piece of the spacelike
graph of a positive constant function defined over the piece of
$\mathscr{H}^{n}(1)$ as time tends to infinity.
