I'll post an answer to spell out all the details.
You have $$G(ω)=\frac{1}{N}E\left[{\rm Tr}\frac{1}{Iω−J}\right]=\frac{1}{N}E\left[\sum_\lambda\frac{1}{ω−\lambda}\right]$$
This can be written as $$G(ω)=\frac{1}{N}E\left[\int d^2z \sum_\lambda\frac{\delta(z-\lambda)}{ω−z}\right],$$ where $\delta$ is the delta-function/distribution in the complex plane.
Now define $$\rho(z)=\frac{1}{N}E\left[\sum_\lambda\delta(z-\lambda)\right]$$, which is the spectral density. Then it follows that $$G(\omega)=\int d^2z \frac{\rho(z)}{\omega-z}$$