The KiDS Strongly lensed QUAsar Detection project (KiDS-SQuaD) aims at finding as many previously undiscovered gravitational lensed quasars as possible in the Kilo Degree Survey. This is the second paper of this series where we present a new, automatic object classification method based on machine learning technique. The main goal of this paper is to build a catalogue of bright extragalactic objects (galaxies and quasars), from the KiDS Data Release 4, with a minimum stellar contamination, preserving the completeness as much as possible, to then apply morphological methods to select reliable gravitationally lensed (GL) quasar candidates. After testing some of the most used machine learning algorithms, decision trees based classifiers, we decided to use CatBoost, that was specifically trained with the aim of creating a sample of extragalactic sources as clean as possible from stars. We discuss the input data, define the training sample for the classifier, give quantitative estimates of its performances, and finally describe the validation results with Gaia DR2, AllWISE, and GAMA catalogues. We have built and make available to the scientific community the KiDS Bright EXtraGalactic Objects catalogue (KiDS-BEXGO), specifically created to find gravitational lenses. This is made of $approx6$ millions of sources classified as quasars ($approx 200,000$) and galaxies ($approx 5.7$M), up to $r<22^m$. From this catalog we selected Multiplets: close pairs of quasars or galaxies surrounded by at least one quasar, presenting the 12 most reliable gravitationally lensed quasar candidates, to demonstrate the potential of the catalogue, which will be further explored in a forthcoming paper. We compared our search to the previous one, presented in the first paper from this series, showing that employing a machine learning method decreases the stars-contaminators within the GL candidates.
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