Aims: In this work, we aim to provide a reliable list of gravitational lens candidates based on a search performed over the entire Gaia Data Release 2 (Gaia DR2). We also aim to show that the astrometric and photometric information coming from the Gaia satellite yield sufficient insights for supervised learning methods to automatically identify strong gravitational lens candidates with an efficiency that is comparable to methods based on image processing. Methods: We simulated 106 623 188 lens systems composed of more than two images, based on a regular grid of parameters characterizing a non-singular isothermal ellipsoid lens model in the presence of an external shear. These simulations are used as an input for training and testing our supervised learning models consisting of extremely randomized trees (ERTs). These trees are finally used to assign to each of the 2 129 659 clusters of celestial objects extracted from the Gaia DR2 a discriminant value that reflects the ability of our simulations to match the observed relative positions and fluxes from each cluster. Once complemented with additional constraints, these discriminant values allow us to identify strong gravitational lens candidates out of the list of clusters. Results: We report the discovery of 15 new quadruply-imaged lens candidates with angular separations of less than 6″ and assess the performance of our approach by recovering 12 of the 13 known quadruply-imaged systems with all their components detected in Gaia DR2 with a misclassification rate of fortuitous clusters of stars as lens systems that is below 1%. Similarly, the identification capability of our method regarding quadruply-imaged systems where three images are detected in Gaia DR2 is assessed by recovering 10 of the 13 known quadruply-imaged systems having one of their constituting images discarded. The associated misclassification rate varies between 5.83% and 20%, depending on the image we decided to remove.The catalogue of clusters of Gaia DR2 sources from Gaia GraL is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/622/A165< Réduire