We study the quantum correlation between the signal and pump modes in forward three-wave mixing in media with an intensity-dependent index of refraction, by adopting the semiclassical and the full-quantum theories. Strong positive correlation exists between the two signal modes, but there is strong anti-correlation between the signal modes and the pump. In the semiclassical model, the phase mismatched cases are investigated. In the full-quantum analysis, the effects of the pump quantization and losses are considered.