  In this paper, we introduce notions of $\alpha$-planes in $5$D complex
Heisenberg group and the twistor space as the moduli space of all
$\alpha$-planes. So we can define an anti-self-dual (ASD) connection as a
connection flat over all $\alpha$-planes. This geometric approach allows us to
establish Penrose-Ward correspondence between ASD connections over $5$D complex
Heisenberg group and a class of holomorphic vector bundles on the twistor
space. By Atiyah-Ward ans\"{a}tz, we also construct a family of ASD connections
on $5$D complex Heisenberg group. When restricted to $5$D real Heisenberg
group, the flat model of $5$D contact manifolds, an ASD connection satisfies
the horizontal part of the contact instanton equation introduced by physicists.
