We investigate the fluctuations in the number of integral lattice points on the Heisenberg groups which lie inside a Cygan-Kor{á}nyi norm ball of large radius. Let $\mathcal{E}_{q}(x)=\big|\mathbb{Z}^{2q+1}\capδ_{x}\mathcal{B}\big|-\textit{vol}\big(\mathcal{B}\big)x^{2q+2}$ denote the error term which occurs for this lattice point counting problem on the Heisenberg group $\mathbb{H}_{q}$, where $\mathcal{B}$ is the unit ball in the Cygan-Kor{á}nyi norm and $δ_{x}$ is the Heisenberg-dilation by $x>0$. For $q\geq3$ we consider the suitably normalized error term $\mathcal{E}_{q}(x)/x^{2q-1}$, and prove it has a limiting value distribution which is absolutely continuous with respect to the Lebesgue measure. We show that the defining density for this distribution, denoted by $\mathcal{P}_{q}(α)$, can be extended to the whole complex plane $\mathbb{C}$ as an entire function of $α$ and satisfies for any non-negative integer $j\geq0$ and any $α\in\mathbb{R}$, $|α|>α_{q,j}$, the bound: \begin{equation*} \begin{split} \big|\mathcal{P}^{(j)}_{q}(α)\big|\leq\exp{\Big(-|α|^{4-β/\log\log{|α|}}\Big)} {split} {equation*} where $β>0$ is an absolute constant. In addition, we give an explicit formula for the $j$-th integral moment of the density $\mathcal{P}_{q}(α)$ for any integer $j\geq1$.