The study of the intersection patterns of two relevant geometric objects provides in some cases new combinatorial characterizations as well as interesting applications. In this work we determine the full list of the possible intersection sizes between a non-singular Hermitian variety H and an irreducible quadric Q in PG(3, q^2), q odd, under the assumption that H and Q share a tangent plane at a common point. We also characterize in geometric terms the configurations attaining the maximum or the minimum value: they correspond to permutable polarities.

Quasi-Hermitian varieties V in PG (r,q^2) are combinatorial generalizations of the (nondegenerate) Hermitian variety H(r,q^2) so that V and H(r,q^2) have the same size and the same intersection numbers with hyperplanes. In this paper, we construct a new family of quasi-Hermitian varieties. The isomorphism problem for the associated strongly regular graphs is discussed for r=2.