Here we apply a technique selecting quasar candidates purely as sources with zero proper motions in the Gaia data release 2 (DR2). We demonstrate that this approach is highly efficient toward high Galactic latitudes with ≲25% contamination from stellar sources. Such a selection technique offers very pure sample completeness, since all cosmological point sources are selected regardless of their intrinsic spectral properties within the limiting magnitude of Gaia. We carry out a pilot-study, defining a sample compiled by including all Gaia-DR2 sources within one degree of the north Galactic pole (NGP) selected to have proper motions consistent with zero within 2σ uncertainty. By cross-matching the sample to the optical Sloan Digital Sky Survey (SDSS) and the mid-infrared (MIR) AllWISE photometric catalogues, we investigate the colours of each of our sources. We determine the efficiency of our selection by comparison with previously spectroscopically confirmed quasars. The majority of the zero-proper-motion sources selected here have optical to MIR colours consistent with known quasars. The remaining population may be contaminating stellar sources, but some may also be quasars with colours similar to stars. Spectroscopic follow-up of the zero-proper-motion sources is needed to unveil such a hitherto hidden quasar population. This approach has the potential to allow substantial progress on many important questions concerning quasars, such as determining the fraction of dust-obscured quasars, the fraction of broad absorption line (BAL) quasars, and the metallicity distribution of damped Lyman-α absorbers. The technique could also potentially reveal new types of quasars or even new classes of cosmological point sources.