Let us consider the Poincaré Upper Half-Plane $H=\{x+iy| x,y \in \textbf{R}, y>0\}$ with arc length element $ds^2=y^{-2}(dx^2+dy^2)$.
We know that the generalized formula for the Laplacian is $\Delta f=|G|^{-1/2}\frac{\partial}{\partial u_i}\left( |G|^{1/2}g^{ij}\frac{\partial f}{\partial u_j} \right)$ where $u$ is the new coordinate, $G=A^TA$, $A=\frac{\partial (u, v)}{\partial (x, y)}$ which is Jacobian of fractional linear transformation $w=u+iv=\frac{az+b}{cz+d}$ of $SL(2,\textbf{R})$.
I have proved that $\det(A)=|dw/dz|^2=|cz+d|^{-4}$ (and that $v=y|cz+d|^{-2}$ if this is relevant).
Using this I can easily find that $|G|^{1/2}=\det(A)=|cz+d|^{-4}$, but from here how should I proceed forward? I think I have to find $g^{ij}$, but since I do not know the explicit form of $G$, I think I'm stuck.
The Laplacian should appear as $\Delta_H=y^2 \left( \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} \right)$.
Additional question: In a similar way or another, can I prove that the area element is $d\mu=y^{-2}dx\,dy$?
Let us compute the Laplacian from the formula $$\tag{1} \frac{1}{\sqrt{|G|}}\partial_i( g^{ij}\sqrt{|G|}\partial_j).$$ The matrix $g_{ij}$ is $$ (g_{ij})=\begin{bmatrix} y^{-2} & 0 \\ 0 & y^{-2}\end{bmatrix}, $$ so $\sqrt{|G|}=\sqrt{\det(g_{ij})}=y^{-2}$. The inverse matrix is $$ (g^{ij})=\begin{bmatrix} y^{2} & 0 \\ 0 & y^{2}\end{bmatrix}$$ Thus, (1) reduces to $$ \frac{1}{y^{-2}}\partial_x(y^2y^{-2}\partial_x) + \frac{1}{y^{-2}}\partial_y(y^2y^{-2}\partial_y), $$ which is clearly equal to $y^2(\partial^2_x+\partial^2_y)$.