Euclid Quick Data Release (Q1): XII. Quenching precedes bulge formation in dense environments but follows it in the field The well-known bimodality between star-forming discs and quiescent spheroids requires the existence of two main processes: galaxy quenching, causing the strong reduction of star formation, and morphological transformation, causing the transition from disc-dominated structures to bulge-dominated ones. In this paper, we aim to understand the link between these two processes and their relation with the stellar mass of galaxies and their local environment. Taking advantage of the first data released by the Euclid Collaboration, covering more than 60 deg2 with space-based imaging and photometry, we analyse a mass-complete sample of nearly one million galaxies in the range 0.25 < z < 1 with M* > 109.5 M⊙, using a combination of photometric and spectroscopic redshifts. We divide the sample into four sub-populations of galaxies, based on their star-formation activity (star-forming and quiescent) and morphology (disc-dominated and bulge-dominated). We then analyse the physical properties of these populations and their relative abundances in the stellar mass versus local density plane. Together with confirming the passivity-density relation and the morphology-density relation, we find that quiescent discy galaxies are more abundant in the low-mass regime of high-density environment where log10(1 + δ) > 1.3. At the same time, star-forming bulge-dominated galaxies are more common in field regions with log10(1 + δ) < 0.8, preferentially at high masses. Building on these results and interpreting them through comparison with simulations, we propose a scenario where the evolution of galaxies in the field significantly differs from that in higher-density environments. The morphological transformation in the majority of field galaxies takes place before the onset of quenching and is mainly driven by secular processes taking place within the main sequence, leading to the formation of star-forming bulge-dominated galaxies as intermediate-stage galaxies. Conversely, quenching of star formation precedes morphological transformation for most galaxies in higher-density environments. This causes the formation of quiescent disc-dominated galaxies before their transition into bulge-dominated ones. 711 EDP Sciences application/pdf