  The $\mathcal{Q}$-graph of a graph $G$, denoted by $\mathcal{Q}(G)$, is the
graph derived from $G$ by plugging a new vertex to each edge of $G$ and adding
a new edge between two new vertices which lie on adjacent edges of $G$. In this
paper, we consider to study the existence of the signless Laplacian perfect
state transfer and signless Laplacian pretty good state transfer in
$\mathcal{Q}$-graphs of graphs. We show that, if all the signless Laplacian
eigenvalues of a regular graph $G$ are integers, then the $\mathcal{Q}$-graph
of $G$ has no signless Laplacian perfect state transfer. We also give a
sufficient condition that the $\mathcal{Q}$-graph of a regular graph has
signless Laplacian pretty good state transfer when $G$ has signless Laplacian
perfect state transfer between two specific vertices.
