Measurements of integrated and differential cross-sections for electroweak W$^{±}$Z production in association with two jets (W$^{±}$Zjj) in proton-proton collisions are presented. The data collected by the ATLAS detector at the Large Hadron Collider from 2015 to 2018 at a centre-of-mass energy of $ \sqrt{s} $ = 13 TeV are used, corresponding to an integrated luminosity of 140 fb$^{−1}$. The W$^{±}$Zjj candidate events are reconstructed using leptonic decay modes of the gauge bosons. Events containing three identified leptons, either electrons or muons, and two jets are selected. Processes involving pure electroweak W$^{±}$Zjj production at Born level are separated from W$^{±}$Zjj production involving a strong coupling. The measured integrated fiducial cross-section of electroweak W$^{±}$Zjj production per lepton flavour is $ {\sigma}_{WZjj- EW\to {\ell}^{\prime}\nu \mathcal{ll} jj} $ = 0.368 ± 0.037 (stat.) ± 0.059 (syst.) ± 0.003 (lumi.) fb, where ℓ and ℓ′ are either an electron or a muon. Respective cross-sections of electroweak and strong W$^{±}$Zjj production are measured separately for events with exactly two jets or with more than two jets, and in three bins of the invariant mass of the two jets. The inclusive W$^{±}$Zjj production cross-section, without separating electroweak and strong production, is also measured to be $ {\sigma}_{WZjj\to {\ell}^{\prime}\nu \mathcal{ll} jj} $ = 1.462 ± 0.063 (stat.) ± 0.118 (syst.) ± 0.012 (lumi.) fb, per lepton flavour. The inclusive W$^{±}$Zjj production cross-section is measured differentially for several kinematic observables. Finally, the measurements are used to constrain anomalous quartic gauge couplings by extracting 95% confidence level intervals on dimension-8 operators.[graphic not available: see fulltext]