A Mobius transformation is a map $$f(x)=\frac{rx+s}{tx+u}$$ where $ru-st \neq 0$. Suppose we have $f(a)=c, f(b)=d, f(c)=a, f(d)=b$, where $a,b,c,d \in \mathbb{R}$. Then from here, the answer given by Noam, the coefficients of the transformation are $$r=ad-bc, s=bcd-acd+abd-abc, t=a-b+c-d, u=bd-ac$$
Question: How to obtain the coefficients? I used the information above and then solve for $r,s,t,u$, but eventually I end up in a mess.
As was pointed out, you need $u=-r$ since the transformation is self-inverse. So you now have $$f(x)=\frac{rx+s}{tx-r}$$ Now let $$\frac sr=P, \frac tr=Q$$ Now you can use $f(a)=c$ and $f(b)=d$ to solve simultaneously for $P$ and $Q$