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.