I am looking for references on contact structures in the framework of pseudo-riemannian manifolds. For instance on the Lorentz-Minkowski $3$-space (resp. $(n+1)$-space). Denoting by $L$ (resp. $H$) the Lorentz-Minkowski $3$-space (resp. the hyperbolic plane regarded as pseudo-sphere in $L$), it seems to me that $w= udx+vdy-wdz, (x,y,z,u,v,w)$ in $LxH$, defines a contact structure on $LxH$. It is as the classical example of the unit tangent bundle of $3$-dimensional Euclidean space E but replacing $E$ by $L$ and the Euclidean scalar product by the Lorentzian one in $\left<(dx,dy,dz),(u,v,w)\right>$.
Yes, this is a contact structure. The point is that, using the pseudo-Riemannian metric, you can identify its future-directed 'unit' tangent bundle with an open set in the projectivized cotangent bundle. Since the projectivized cotangent bundle of any manifold carries a canonical contact structure, this identification induces a canonical contact structure on the future-directed 'unit' tangent bundle. In your case, $L\times H$ is exactly the future-directed unit tangent bundle. As you note, it corresponds to the unit sphere bundle of a Riemannian manifold (which also inherits a contact structure in exactly the same way).