Looking at Sylow questions on $GL_2(\mathbb F_3)$. we have that $Q$ is the unique $2$-Sylow of $N=SL_2(\mathbb F_3)$. $|Q|=8=2^3$ hence by the classification of groups of order $p^3$, we have 5 possibilies for $Q$: $\;C_8,\;C_4\times C_2,\;C_2^3,\;D_4,\;Q_8$. But looking at On $GL_2(\mathbb F_3)$, we see that we have only one element of order $2$ in $N$, hence the same holds for $Q$, then we must exclude $C_4\times C_2,\;C_2^3,\;D_4$.
Hence $Q=C_8$ or $Q=Q_8$. In order to exclude the case $Q=C_8$ we note that even though $Q\unlhd N$, $Q$ is not centralized by any $3$-Sylow subgroup of $N$ (there are four $3$-Sylow subgroups of $N$, see again Sylow questions on $GL_2(\mathbb F_3)$.), and so taken $b\in N$ an element of order $3$, we have that the conjugation $q\mapsto q^b$ is a non trivial automorphism of $Q$. How can we show that this automorphism has order $3$?
Direct computations seems don't lead anywhere.
Proving that there exists an automorphism of order $3$, we can argue as follows: if by contradiction $Q=C_8$, then $Aut(C_8)\simeq U(\mathbb Z_8)$ hence $|Aut(C_8)|=4$, thus it can't contain any automorphism of order $3$. Then I can conclude that $Q=Q_8$.
Thank you all