of quantum noise. Here we show that a superconducting qubit strongly coupled to a nonlinear resonator can act as a probe of quantum fluctuations of the intra-resonator field. Building on previous work [M. Boissoneault et al. Phys. Rev. A 85, 022305 (2012)], we derive an effective master equation for the qubit which takes into account squeezing of the resonator field. We show how sidebands in the qubit excitation spectrum that are predicted by this model can reveal information about squeezing and quantum heating. The main results of this paper have already been successfully compared to experimental data [F. R. Ong et al. Phys. Rev. Lett. 110, 047001 (2013)] and we present here the details of the derivations.