In July of 2019, the IceCube experiment detected a high-energy neutrino from the direction of the powerful blazar PKS 1502+106. We perform multi-wavelength and multi-messenger modeling of this source, using a fully self-consistent one-zone model that includes the contribution of external radiation fields typical of flat-spectrum radio quasars (FSRQs). We identify three different activity states of the blazar: the quiescent state, and two distinct flaring states with hard and soft gamma-ray spectra. We find two hadronic models that can both describe the multi-wavelength emission during all three states: a leptohadronic model with a contribution from photo-hadronic processes to X-rays and high-energy gamma rays, and a proton synchrotron model, where the emission from keV to 10 GeV comes from proton synchrotron radiation. Both models predict a substantial neutrino flux that is correlated with the gamma-ray and soft X-ray fluxes. Our results are compatible with the detection of a neutrino during the quiescent state, based on event rate statistics. Upon an extensive parameter scan, we conclude that the soft X-ray spectra observed during bright gamma-ray flares strongly suggest a hadronic contribution, which can be interpreted as additional evidence for cosmic ray acceleration in the source independently of neutrino observations. We find that more arguments can be made in favor of the leptohadronic model vis-a-vis the proton synchrotron scenario, such as a lower energetic demand during the quiescent state, and the fact that the same model has also been shown to describe the observation of neutrinos from blazar TXS 0506+056. The leptohadronic model would be disfavored for flaring states of PKS 1502+106 if no IceCube events were found from the direction of the source before 2010, which would require an archival search.