  In this paper, we compute for odd fundamental discriminants $D>1$ the Fourier
expansion of non-holomorphic elliptic Eisenstein series for $\Gamma_0(D)$ with
quadratic nebentypus character $\chi_D$ satisfying a certain plus space
condition. For each genus of $\mathbb{Q}(\sqrt{D})$, we obtain an associated
plus space condition and corresponding Eisenstein series in all positive even
weights. In weight $k=2$, the Fourier coefficients are associated to the
geometry of Hirzebruch--Zagier divisors on Hilbert modular surfaces.
