  The signless Laplacian matrix in graph spectra theory is a remarkable matrix
of graphs, and it is extensively studied by researchers. In 1981, Cvetkovi\'{c}
pointed $12$ directions in further investigations of graph spectra, one of
which is "classifying and ordering graphs". Along with this classic direction,
we pay our attention on the order of the largest eigenvalue of the signless
Laplacian matrix of graphs, which is usually called the $Q$-index of a graph.
Let $\mathbb{G}(m, g)$ (resp. $\mathbb{G}(m, \geq g)$) be the family of
connected graphs on $m$ edges with girth $g$ (resp. no less than $g$), where
$g\ge3$. In this paper, we firstly order the first
$(\lfloor\frac{g}{2}\rfloor+2)$ largest $Q$-indices of graphs in $\mathbb{G}(m,
g)$, where $m\ge 3g\ge 12$. Secondly, we order the first
$(\lfloor\frac{g}{2}\rfloor+3)$ largest $Q$-indices of graphs in $\mathbb{G}(m,
\geq g)$, where $m\ge 3g\ge 12$. As a complement, we give the first five
largest $Q$-indices of graphs in $\mathbb{G}(m, 3)$ with $m\ge 9$. Finally, we
give the order of the first eleven largest $Q$-indices of all connected graphs
with size $m$.
