  We construct explicit families of quasi-hyperbolic and hyperbolic surfaces.
This is based on earlier work of Vojta, and the recent expansion and
generalization of it by the first author. In this paper we further extend it to
the singular case, obtaining results for the surface of cuboids, the
generalized surfaces of cuboids, and other families of Diophantine surfaces of
general type. In particular, we produce explicit families of smooth complete
intersection surfaces of multidegrees $(m_1,\ldots,m_n)$ in $\mathbb{P}^{n+2}$
which are hyperbolic, for any $n \geq 8$ and any degrees $m_i \geq 2$. We also
show similar results for complete intersection surfaces in $\mathbb{P}^{n+2}$
for $n=4,5,6,7$. These families give evidence for [Dem18, Conjecture 0.18] in
the case of surfaces.
