  Let $G$ be a graph of order $n$, and let $q_1(G)\geq q_2(G)\geq\cdots\geq
q_n(G)$ denote the signless Laplacian eigenvalues of $G$. Ashraf and
Tayfeh-Rezaie [Electron. J. Combin. 21 (3) (2014) \#P3.6] showed that
$q_1(G)+q_1(\overline{G})\leq 3n-4$, with equality holding if and only if $G$
or $\overline{G}$ is the star $K_{1,n-1}$. In this paper, we discuss the
following problem: for $n\geq6$, does $q_2(G)+q_2(\overline{G})\leq 2n-5$
always hold? We provide positive answers to this problem for the graphs with
disconnected complements and the bipartite graphs, and determine the graphs
attaining the bound. Moreover, we show that $q_2(G)+q_2(\overline{G})\geq n-2$,
and the extremal graphs are also characterized.
