Aims. Quasar strong gravitational lenses are important tools for putting constraints on the dark matter distribution, dark energy contribution, and the Hubble-Lemaître parameter. We aim to present a new supervised machine learning-based method to identify these lenses in large astrometric surveys. The Gaia Focused Product Release (FPR) GravLens catalogue is designed for the identification of multiply imaged quasars, as it provides astrometry and photometry of all sources in the field of 4.7 million quasars. Methods. Our new approach for automatically identifying four-image lens configurations in large catalogues is based on the eXtreme Gradient Boosting classification algorithm. To train this supervised algorithm, we performed realistic simulations of lenses with four images that account for the statistical distribution of the morphology of the deflecting halos as measured in the EAGLE simulation. We identified the parameters discriminant for the classification and performed two different trainings, namely, with and without distance information. Results. The performances of this method on the simulated data are quite good, with a true positive rate and a true negative rate of about 99.99% and 99.84%, respectively. Our validation of the method on a small set of known quasar lenses demonstrates its efficiency, with 75% of known lenses being correctly identified. We applied our algorithm (both trainings) to more than 0.9 million quadruplets selected from the Gaia FPR GravLens catalogue. We derived a list of 1127 candidates with at least one score larger than 0.75, where each candidate has two scores-one from the model trained with distance information and one from the model trained without distance information-and including 201 very good candidates with both high scores.