Here is a paper extending Pick's Theorem to cover the case where the lattice is hexagonal.
The things that remains a puzzle to me is what is meant by the 'boundary characteristic'. The authors introduce this parameter as the "number of edge extended locally into the exterior of P minus those extended locally into the interior of P", where P is the polygon in question. Can anybody help explain this more thoroughly, ideally with a simple example?
(If you are looking for a solution for triangular lattices, here is a good thread I found: Pick's Theorem on a triangular (or hex) grid)