  In this paper, we consider commuting conjugacy class graph (abbreviated as
CCC-graph) of a finite group $G$ which is a graph with vertex set $\{x^G : x
\in G \setminus Z(G)\}$ (where $x^G$ denotes the conjugacy class containing
$x$) and two distinct vertices $x^G$ and $y^G$ are joined by an edge if there
exist some elements $x'\in x^G$ and $y'\in y^G$ such that they commute. We
compute common neighborhood (signless) Laplacian spectrum and energy of
CCC-graph of finite non-abelian groups whose central quotient is isomorphic to
either $\mathbb{Z}_p \times \mathbb{Z}_p$ (where $p$ is any prime) or the
dihedral group $D_{2n}$ ($n \geq 3$); and determine whether CCC-graphs of these
groups are common neighborhood (signless) Laplacian
hyperenergetic/borderenergetic. As a consequence, we characterize certain
finite non-abelian groups viz. $D_{2n}$, $T_{4n}$, $U_{6n}$, $U_{(n, m)}$,
$SD_{8n}$ and $V_{8n}$ such that their CCC-graphs are common neighborhood
(signless) Laplacian hyperenergetic/borderenergetic.
