  In this paper we introduce the fourth fundamental form for the hypersurfaces
in $H^{n+1}$ and the space-like hypersurfaces in $S_{1}^{n+1}$ and discuss the
conformality of the normal Gauss maps of the hypersurfaces in $H^{n+1}$ and
$S_{1}^{n+1}$. Particularly, we discuss the surfaces with conformal normal
Gauss maps in $H^{3}$ and $S_{1}^{3}$ and prove a duality property. We give the
Weierstrass representation formula for the space-like surfaces in $S_{1}^{3}$
with conformal normal Gauss maps. We also state the similar results for the
time-like surfaces in $S_{1}^{3}.$
