Well, I solved it, and I would like to know if there is anything that can be corrected or improved here. I think that the proof ended up too long, and with too many letters. Surely there is a better way to write it. Alternate solutions are welcome too. Thank you.
Proof: Suppose by contradiction that exists $s \in \Bbb Q$ such that $s^2 = 6$. Then, we have $s = \dfrac{p}{q}$, with $p,q \in \Bbb Z, q \neq 0, \gcd(p,q) = 1$. The standard strategy is to contradict the part about the $\gcd$. We have: $$\begin{align} \left(\frac{p}{q}\right)^2 &= 6 \\ p^2 &= 6q^2 \\ p^2 &= 2(3q^2)\end{align}$$ so $p^2$ is even, and it follows that $p$ is even, and so exists $m \in \Bbb Z$, with $p = 2m$. Proceeding, we have: $$\begin{align} (2m)^2 &= 2(3q^2) \\ 4m^2 &= 2(3q^2) \\ 2m^2 &= 3q^2\end{align}$$ From here, we have that $3q^2$ is even. If $q$ is also even, $\gcd(p,q) \neq 1$ and we're finished. Let's see the case that $q$ is odd. Then exists $\ell \in \Bbb Z$, with $q = 2 \ell + 1$. Proceeding in this case: $$\begin{align} 2m^2 &= 3(2 \ell + 1)^2 \\ 2m^2 &= 3(4 \ell^2 + 4\ell + 1) \\ 2m^2 &= 12 \ell^2 + 12 \ell + 3 \\ 2m^2 &= 2(6 \ell^2 + 6 \ell + 1) + 1 \end{align}$$ which is a contradiction, because the left-hand side is even, and the right-hand side is odd. Therefore $q$ must be even, and $\gcd(p,q) \neq 1$. Hence, there is no rational number whose square is $6$.