Correlation function and Green's function

Question

I have referred to several materials on Green's function but I found those notions pretty confusing. Now what I have tried to do is to calculate the inverse matrix of $G^+(E)=E-H+\mathrm{i}\eta$ where $H$ is a given lattice Hamiltonian in Wannier representation, then the trace of the imaginary part of $G^+(E)$ turns out to be the DOS and my results seem quite good. And I suppose that this is more likely to be a 'one-body' Green's function, while what I actually need is the many-body correlation function $\langle\Omega|c_ic_j^\dagger|\Omega\rangle$ where for simplicity we assume the ground state to be half-filled. And besides also I don't understand what the relation is between $G^+$ and $\langle\Omega|T(\psi(r,t)\psi^\dagger({r',t'})|\Omega\rangle$ since my system does not concern any time evolution.