My guess would be indeed $\tfrac{1}{2} \partial_{yy}$, with $f$ in the domain if and only if:

(a) $f$ is in $C_0(\mathbb{R}_+^2)$;

(b) $f(x, \cdot)$ is $C_0^2(\mathbb{R}_+)$ for each $x$;

(c) $\partial_{yy} f$ is in $C_0(\mathbb{R}_+^2)$;

(d) $f(\cdot, 0)$ is $C_0^2(\mathbb{R}_+)$ and $\partial_{xx} f(x, 0) = \partial_{yy} f(x, 0)$ for all $x$.

I do not know any reference for this. It should be relatively simple to show that any $f$ as above belongs to the domain (by proving convergence of $(P_t f - f) / t$ to $\tfrac{1}{2} \partial_{yy} f$, or, simpler, by using Dynkin's characteristic operator). The converse is always much more problematic.