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 $Q=C_8$ my teacher said 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 $Q$ has an automorphism of order $3$.
I think that by "$Q$ is not centralized by any $3$-Sylow subgroup of $N$", my teacher mean that although $Q\unlhd N$, and thus $Q^g=Q\;\;\forall g\in N$, calling $B$ a $3$-Sylow of $N$, it's not true that $qb=bq\;\;\forall q\in Q,\;\;\forall b\in B$ (first question: how can I see this?), hence we can define a $\psi\in Aut(Q)$ defined by $\psi(q)=q^b$; and the second question is: how can I prove that $\psi$ has order $3$ (the order of an automorphism is defined as the minmum $n\in\mathbb N$ s.t. $\psi^n=id_Q$)? And if $\psi$ is not the right automorphism, (third question) how can I define an automorphism of $Q$ of order $3$?
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
Have you thought about exploiting the structure of $PGL(2,\mathbb F_3) := GL(2,\mathbb F_3)/(\pm 1)$? This has order $24$, and acts faithfully on the $4$ points of the projective line over $\mathbb F_3$. Its subgroup $PSL(2,\mathbb F_3)$ of index $2$ thus has order $12$. Given what I've just said, you can probably figure out what these groups are.
This will make it pretty straightforward to answer your questions (and any similar ones).
Alternatively, you can just write down some elements in $GL(2,\mathbb F_3)$ of $2$-power order, and an element of order $3$, and see if they commute. (These are just $2\times 2$ matrices, so it's not very hard to compute with them directly.)