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The origin of the radio emission in radio-quiet quasars (RQQs) remains unclear. Radio emission may be produced by a scaled-down version of the relativistic jets observed in radio-loud (RL) AGN, an AGN-driven wind, the accretion disc corona, AGN photon-ionization of ambient gas (free–free emission), or star formation (SF). Here, we report a pilot study, part of a radio survey (‘PG-RQS’) aiming at exploring the spectral distributions of the 71 Palomar–Green (PG) RQQs: high angular resolution observations (∼50 mas) at 45 GHz (7 mm) with the Karl G. Jansky Very Large Array of 15 sources. Sub-mJy radio cores are detected in 13 sources on a typical scale of ∼100 pc, which excludes significant contribution from galaxy-scale SF. For 9 sources the 45-GHz luminosity is above the lower frequency (∼1–10 GHz) spectral extrapolation, indicating the emergence of an additional flatter-spectrum compact component at high frequencies. The X-ray luminosity and black hole (BH) mass, correlate more tightly with the 45-GHz luminosity than the 5-GHz. The 45 GHz-based radio-loudness increases with decreasing Eddington ratio and increasing BH mass MBH. These results suggest that the 45-GHz emission from PG RQQs nuclei originates from the innermost region of the core, probably from the accretion disc corona. Increasing contributions to 45-GHz emission from a jet at higher MBH and lower Eddington ratios and from a disc wind at large Eddington ratios are still consistent with our results. Future full radio spectral coverage of the sample will help us investigating the different physical mechanisms in place in RQQ cores.