  Let $q(G)$ denote the $Q$-index of a graph $G$, which is the largest signless
Laplacian eigenvalue of $G$. We prove best possible upper bounds of $q(G)$ and
best possible lower bounds of $q(\overline{G})$ for a connected graph $G$ to be
$k$-connected and maximally connected, respectively. Similar upper bounds of
$q(G)$ and lower bounds of $q(\overline{G})$ to assure $G$ to be
super-connected are also obtained. Upper bounds of $q(G)$ and lower bounds of
$q(\overline{G})$ to assure a connected triangle-free graph $G$ to be
$k$-connected, maximally connected and super-connected are also respectively
investigated.
