Classical Cepheids (DCEPs) represent a fundamental tool to calibrate the extragalactic distance scale. However, they are also powerful stellar population tracers, in the context of Galactic studies. The forthcoming Data Release 3 (DR3) of the Gaia mission will allow us to study with unprecedented detail the structure, the dynamics and the chemical properties of the Galactic disc, and in particular of the spiral arms, where most Galactic DCEPs reside. In this paper we aim at quantifying the metallicity dependence of the Galactic DCEPs Period-Wesenheit (PWZ) relation in the Gaia bands. We adopt a sample of 499 DCEPs with metal abundances from high-resolution spectroscopy, in conjunction with Gaia Early Data Release 3 parallaxes and photometry to calibrate a PWZ relation in the Gaia bands. We find a significant metallicity term, of the order of -0.5mag/dex, which is larger than the values measured in the NIR bands by different authors. Our best PWZ relation is W=(-5.988+/-0.018)-(3.176+/-0.044)(logP-1.0)-(0.520+/-0.090)[Fe/H]. We validate our PWZ relations by using the distance to the Large Magellanic Cloud as a benchmark, finding a very good agreement with the geometric distance provided by eclipsing binaries. As an additional test, we evaluate the metallicity gradient of the young Galactic disc, finding -0.0527+/-0.0022dex/kpc, in very good agreement with previous results.