There are counterexamples for $Q=J$. 

Here is a binary $6\times7$ binary matrix $M$ that belongs to an orbit of $\mathcal{L}_J$ with period $3$:

$\begin{matrix}
1&1&1&0&0&0&0\\
1&1&0&1&1&0&0\\
0&0&1&1&0&1&0\\
0&0&1&0&1&0&1\\
1&0&0&1&0&1&1\\
0&1&0&0&1&1&1\\
\end{matrix}$

Here is another with period $4$:

$\begin{matrix}
1&1&0&0&0&0&0\\
1&0&1&1&0&0&0\\
0&1&1&0&1&0&0\\
0&1&0&1&0&1&1\\
1&0&0&0&1&1&0\\
0&0&0&1&1&1&0\\
\end{matrix}$

and another with period $6$:

$\begin{matrix}
1&1&1&0&0&0&0\\
1&1&0&1&1&0&0\\
0&0&1&1&0&1&0\\
0&0&1&1&0&0&1\\
1&0&0&0&0&1&1\\
0&1&0&0&0&1&1\\
\end{matrix}$

On the whole set of $7\times 7$ binary matrices, $\mathcal{L}_J$ gets
$$
\begin{array}{r l}
     326\,166&\text{fixed matrices}\\
 86\,146\,036&\text{distinct orbits of length }2\\
           94&\text{distinct orbits of length }3\\
       5\,400&\text{distinct orbits of length }4\\
            8&\text{distinct orbits of length }5\\
          196&\text{distinct orbits of length }6\\
\end{array}
$$