So I got stuck trying to find a conformal mapping from Q onto the upper half plane set D = {z: Im(z) > 0}.
I've ultimately arrived at the conclusion that maybe $w = z^3$ could work as the first step but I just can't figure out what it would transform Q into. I understand that I can say that $z = re^{it}$, where $r = 1$, and then $w = r^3 e^{i \cdot 3t}$, but it doesn't really help.
The idea is to map Q to $\mathbb{C}\setminus(-\infty, \alpha]$ for $\alpha\in\mathbb{R}$. This can be realized by using the möbiustranform
T(z)=$\frac{iz+1}{z+i}$.
This möbiustranform should map Q onto $\mathbb{C}\setminus(-\infty,1]$.
Now use the translation $z\mapsto z-1,25$ to get to $\mathbb{C}\setminus(-\infty,-0,25]$. From there you use koebe's function to get to the unit disc and the cayley transform to finally get to the upper half plane.
Note how you get T: When you think in the RS. You would wish to have a möbiustransform T that "rotates the RS 90 degrees towards you", explicitly it should map:
$-i\mapsto\infty$; $-1\mapsto -1$; $i\mapsto 0$; $1\mapsto 1$;
Using these information you get the T above.