There are counterexamples for $Q=J$.
Here is a binary $6\times7$ binary matrix $M$ that belongs to an orbit of $L_J$$\mathcal{L}_J$ with period $3$:
$\begin{matrix} 1&1&1&0&0&0&0\\ 1&1&0&1&1&0&0\\ 0&0&1&1&0&1&0\\ 0&0&1&0&1&0&1\\ 1&0&0&1&0&1&1\\ 0&1&0&0&1&1&1\\ \end{matrix}$
Here is another with period $4$:
$\begin{matrix} 1&1&0&0&0&0&0\\ 1&0&1&1&0&0&0\\ 0&1&1&0&1&0&0\\ 0&1&0&1&0&1&1\\ 1&0&0&0&1&1&0\\ 0&0&0&1&1&1&0\\ \end{matrix}$
and another with period $6$:
$\begin{matrix} 1&1&1&0&0&0&0\\ 1&1&0&1&1&0&0\\ 0&0&1&1&0&1&0\\ 0&0&1&1&0&0&1\\ 1&0&0&0&0&1&1\\ 0&1&0&0&0&1&1\\ \end{matrix}$
On the whole set of $7\times 7$ binary matrices, $\mathcal{L}_J$ gets $$ \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} $$