  Conformal qc geometry of spherical qc manifolds are investigated. We
construct the qc Yamabe operators on qc manifolds, which are covariant under
the conformal qc transformations. A qc manifold is scalar positive, negative or
vanishing if and only if its qc Yamabe invariant is positive, negative or zero,
respectively. On a scalar positive spherical qc manifold, we can construct the
Green function of the qc Yamabe operator, which can be applied to construct a
conformally invariant tensor. It becomes a spherical qc metric if the qc
positive mass conjecture is true. Conformal qc geometry of spherical qc
manifolds can be applied to study convex cocompact subgroups of ${\rm
Sp}(n+1,1).$ On a spherical qc manifold constructed from such a discrete
subgroup, we construct a spherical qc metric of Nayatani type. As a corollary,
we prove that such a spherical qc manifold is scalar positive, negative or
vanishing if and only if the Poincar\'e critical exponent of the discrete
subgroup is less than, greater than or equal to $2n+2$, respectively.
