The problem of counting integer lattice points inside, on, and near large dilates of convex surfaces is a classic and time-honored problem in number theory and related areas. Given a non-empty convex set B ⊂ Rn, one expects that the number of lattice points in RB is well approximated by the volume. We are interested in studying the error in this approximation in a particular setting. In particular, we estab- lish an error estimate for the number of points in a ball of large radius R described by the natural radial and Heisenberg-homogeneous norms on the Heisenberg groups given by Nα,A((z, t)) = |z|α + A|t|α/21/α, for α ≥ 2. Our method of bounding the error term involves obtain- ing decay estimates on the Euclidean Fourier transform of these balls. We comment on the extend to which our result is sharp and make comparison with an analogue of our method to some Euclidean lattice point counting results. This is joint work with Rahul Garg and Amos Nevo of the Israel Institute of Technology