  In 2018, Chen and Godsil proposed the concept of Laplacian perfect pair state
transfer which is a brilliant generalization of Laplacian perfect state
transfer. In this paper, we study the existence of Laplacian perfect pair state
transfer in the Q-graph of an $r$-regular graph for $r\ge2$. We prove that the
Q-graph of an $r$-regular graph does not have Laplacian perfect pair state
transfer when $r+1$ is prime or a power of $2$. We also give a sufficient
condition for Q-graph to have Laplacian pretty good pair state transfer.
