= Solution
Let $X=(x,y)$ be the integration variable and $P=(\xi,\eta)$ the interior source point. <Green's second identity>, obtained from <Green's theorem> by applying the <divergence theorem> to $u\nabla G-G\nabla u$, is
$$
\int_D(u\Delta G-G\Delta u)\,dA
=\int_{\partial D}\left(u\frac{\partial G}{\partial n}-G\frac{\partial u}{\partial n}\right)ds.
$$
The outward normal is used throughout. To obtain the prescribed representation with its positive volume term, choose
$$
\boxed{\Delta_XG(X,P)=\delta(X-P),\qquad
G=0\text{ on }\partial D_1,\qquad
\partial_nG=0\text{ on }\partial D_2.}
$$
Indeed, substituting $\Delta u=f$ makes the left side $u(P)-\int_DGf$. On the Dirichlet portion, $u=g$ and $G=0$ leave only $g\partial_nG$. On the Neumann portion, both normal derivatives vanish. Rearranging gives exactly the desired boundary-plus-volume representation. The distributional identity can equivalently be justified by excluding a small circle around $P$ and using the logarithmic singularity there.
For the quarter-plane, the <fundamental solution of the Laplace equation> with this sign convention is $\Phi(X,P)=(2\pi)^{-1}\log|X-P|$, since $\Delta\Phi=\delta_P$. The <method of images> uses an equal-sign reflection across the Neumann boundary $x=0$ and an opposite-sign reflection across the Dirichlet boundary $y=0$. Hence
$$
G(X,P)=\frac1{2\pi}\bigl[\log|X-(\xi,\eta)|+\log|X-(-\xi,\eta)|
-\log|X-(\xi,-\eta)|-\log|X-(-\xi,-\eta)|\bigr].
$$
Equivalently,
$$
\boxed{G(x,y,\xi,\eta)=\frac1{4\pi}\log
\frac{\bigl((x-\xi)^2+(y-\eta)^2\bigr)\bigl((x+\xi)^2+(y-\eta)^2\bigr)}
{\bigl((x-\xi)^2+(y+\eta)^2\bigr)\bigl((x+\xi)^2+(y+\eta)^2\bigr)}.}
$$
Only $(\xi,\eta)$ lies inside the domain, so the interior <Laplacian> has exactly the required unit point source. At $y=0$ numerator and denominator agree, giving $G=0$. The formula is even in $x$, giving $\partial_xG=0$ at $x=0$, and therefore zero outward normal derivative there.
On the Dirichlet boundary $y=0$, the outward normal is $-\partial_y$, and differentiation gives the useful check
$$
\frac{\partial G}{\partial n}(x,0,\xi,\eta)
=\frac\eta\pi\left[\frac1{(x-\xi)^2+\eta^2}+\frac1{(x+\xi)^2+\eta^2}\right]>0.
$$
This is the <mixed Dirichlet-Neumann Green function in a quadrant>. The image solution tends to zero at spatial infinity; on the unbounded domain the representation uses data and solutions for which the <boundary term at infinity in Green's second identity> vanishes.
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