To map a square to a unit disk, I consider the Möbius transformation because the Möbius transformation maps circles and lines to circles and lines. And because the Möbius transformation can be determined uniquely by three points, I assign $z_1=-1+i$ to $\frac{-1+i}{\sqrt{2}}$ and $z_2=i$ to $i$ and $z_3=1+i$ to $\frac{1+i}{\sqrt{2}}$ . And, I think by completing the four fragments of a square in a similar way, I can obtain the transformation.
However, the computation is so complicated. Is there another simpler method? Thank you!
You can't do this by a Mobius transformation.
But you can use elliptic functions. If your square is the unit square with opposite vertices at $0$ and $1+i$, consider the lattice generated by $2$ and $2i$. The Weierstrass $\wp$-function associated to that lattice will map the interior of the square conformally to a half plane. Then a Mobius transformation will take that to the disc.