  This paper deals with the existence and multiplicity of nontrivial solutions
for $(p, Q)$-Laplace equations with the Stein-Weiss reaction under critical
exponential nonlinearity in the Heisenberg group $\mathbb{H}^N$. In addition, a
weight function and two positive parameters have also been included in the
nonlinearity. The developed analysis is significantly influenced by these two
parameters. Further, the mountain pass theorem, the Ekeland variational
principle, the Trudinger-Moser inequality, the doubly weighted
Hardy-Littlewood-Sobolev inequality and a completely new Br\'ezis-Lieb type
lemma for Choquard nonlinearity play key roles in our proofs.
