We present the VST ATLAS Quasar Survey, consisting of ~1229 000 quasar (QSO) candidates with 16 < g < 22.5 over ~4700 deg2. The catalogue is based on VST ATLAS+NEOWISE imaging surveys and aims to reach a QSO sky density of 130 deg-2 for $z$ < 2.2 and ~30 deg-2 for $z$ > 2.2. To guide our selection, we use X-ray/UV/optical/MIR data in the extended William Herschel Deep Field (WHDF) where we find a g < 22.5 broad-line QSO density of 269 ± 67 deg-2, roughly consistent with the expected ~196 deg-2. We find that ~25 per cent of our QSOs are morphologically classed as optically extended. Overall, we find that in these deep data, MIR, UV, and X-ray selections are ~70-90 per cent complete while X-ray suffers less contamination than MIR and UV. MIR is however more sensitive than X-ray or UV to $z$ > 2.2 QSOs at g < 22.5 and the $S_X(0.5-10\, {\rm keV})\gt 1\times 10^{-14}$ ergs cm-2 s-1 limit of eROSITA. We adjust the selection criteria from our previous 2QDES pilot survey and prioritize VST ATLAS candidates that show both UV and MIR excess, also selecting candidates initially classified as extended. We test our selections using data from DESI (which will be released in DR1) and 2dF to estimate the efficiency and completeness, and we use ANNz2 to determine photometric redshifts. Applying over the ~4700 deg2 ATLAS area gives us $\sim 917\,000\, z\lt 2.2$ QSO candidates of which 472 000 are likely to be $z$ < 2.2 QSOs, implying a sky density of ~100 deg-2, which our WHDF analysis suggests will rise to at least 130 deg-2 when eROSITA X-ray candidates are included. At $z$ > 2.2, we find ~310() 000 candidates, of which 169 000 are likely to be QSOs for a sky density of ~36 deg-2.