Two-quantum two-dimensional electronic spectroscopy (2Q 2D ES) may provide a measure of electron-correlation energies in molecules. Attempts to obtain this profound but elusive signal have relied on experimental implementations using femtosecond laser pulses, which induce an overwhelming background signal of nonresonant response. Here we explore theoretically the signatures of electron correlation in coherent 2Q 2D ES measurements that use spectrally incoherent light, I(4) 2Q 2D ES. One can use such fields to suppress nonresonant response, and therefore this method may better isolate the desired signature of electron correlation. Using an appropriate treatment of the multilevel Bloch electronic system, we find that I(4) 2Q 2D ES presents an opportunity to measure electron-correlation energies in molecules.