For a graph $G$, the signless Laplacian matrix $Q(G)$ defined as $Q(G) = D(G) + A(G)$, where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ the diagonal matrix whose main entries are the degrees of the vertices in $G$. The $Q$-spectrum of $G$ is that of $Q(G)$. In the present paper, we are interested in the minimum values of the second largest signless Laplacian eigenvalue $q_2(G)$ of a connected graph $G$. We find the five smallest values of $q_2(G)$ over the set of connected graphs $G$ with given order $n$. We also characterize the corresponding extremal graphs.