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Since July 2014, the Gaia space mission has been continuously scanning the sky and observing the extragalactic Universe with unprecedented spatial resolution in the optical domain ($\sim$ 180 mas by the end of the mission). Gaia provides an opportunity to study the morphology of the galaxies of the local Universe (z<0.45) with much higher resolution than has ever been attained from the ground. It also allows us to provide the first morphological all-sky space catalogue of nearby galaxies and galaxies that host quasars in the visible spectrum. We present the Data Processing and Analysis Consortium CU4-Surface Brightness Profile fitting pipeline, which aims to recover the light profile of nearby galaxies and galaxies hosting quasars. The pipeline uses a direct model based on the Radon transform to measure the two-dimensional surface brightness profile of the extended sources. It simulates a large set of 2D light profiles and iteratively looks for the one that best reproduces the 1D observations by means of a Bayesian exploration of the parameters space. We also present our method for setting up the input lists of galaxies and quasars to be processed. We successfully analysed 1\,103\,691 known quasars and detected a host galaxy around 64\,498 of them ($\sim$6\%). We publish the surface brightness profiles of the host for a subset of 15\,867 quasars with robust solutions. The distribution of the Sérsic index describing the light profile of the host galaxies peaks at $\sim$ 0.8 with a mean value of $\sim$ 1.9, indicating that these galaxies hosting a quasar are consistent with disc-like galaxies. The pipeline also analysed 940\,887 galaxies with both a \sersic and a de Vaucouleurs profile and derived robust solutions for 914\,837 of them. The distribution of the Sérsic indices confirms that \gaia mostly detects elliptical galaxies and that very few discs are measured.