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Let $G$ be an $n$-vertex graph and $Q(G)$ be its signless Laplacian matrix.The $Q$-walk matrix of $G$, denoted by $W_Q(G)$, is$[e,Q(G)e,\ldots,Q^{n-1}(G)e]$, where $e$ is the all-one vector. Let $G\circP_m$ be the graph obtained from $G$ and $n$ copies of the path $P_m$ byidentifying the $i$-th vertex of $G$ with an endvertex of the $i$-th copy of$P_m$ for each $i$. We prove that, $$\det W_Q(G\circ P_m)=\pm (\detQ(G))^{m-1}(\det W_Q(G))^m$$ holds for any $m\ge 2$. This gives a signlessLaplacian counterpart of the following recently established identity [17]:$$\det W_A(G\circ P_m)=\pm (\det A(G))^{\lfloor\frac{m}{2}\rfloor}(\detW_A(G))^m,$$ where $A(G)$ is the adjacency matrix of $G$ and$W_A(G)=[e,A(G)e,\ldots,A^{n-1}(G)e]$. We also propose a conjecture to unifythe above two equalities.

On the determinant of the $Q$-walk matrix of rooted product with a path