Prove that the polynomial $P=2X+1 \in \Bbb Z_4[X]$ has an inverse element. 
Prove that the polynomial $P=2X+1 \in \Bbb Z_4[X]$ has an inverse element. What happens if we consider $P$ as an element of $\Bbb Q [X]$?

If $P \in \Bbb Z_4[X]$, then any $Q=kX+1$ where $k \equiv 2 \pmod{4}$ will work as an inverse right? I suppose that the result will not change in $\Bbb Q [X]$ as the coefficients have a multiplicative inverses except $0$?

You are correct regarding $Q=kX+1$ being an inverse whenever $k\equiv2\mod 4$.
However, the result changes drastically if we move to $\mathbb Q[X]$ as this is a polynomial ring of an integral domain. You should convince yourself that for $R$ and integral domain and $f,g\in R[X]$ we have
$$
\deg fg=\deg f+\deg g
$$
making invertibility for any non-constant polynomial impossible.
