Machine learning methods are well established in the classification of quasars (QSOs). However, the advent of light-curve observations adds a great amount of complexity to the problem. Our goal is to use the Zwicky Transient Facility (ZTF) to create a catalog of QSOs. We process the ZTF DR20 light curves with a transformer artificial neural network and combine different surveys with extreme gradient boosting. Based on ZTFg-band and Wide-field Infrared Survey Explorer (WISE) observations, we find 4,849,574 objects classified as QSOs with confidence higher than 90% (QZO). We robustly classify objects fainter than the 5σsignal-to-noise ratio (SNR) limit atg= 20.8 by requiringg<n_obs/80+20.375. For 33% of QZO objects, with available WISE data, we publish redshifts with estimated error Δz/(1+z)=0.14. We find that ZTF classification is superior to the Pan-STARRS static bands, and on par with WISE and Gaia measurements, but the light curves provide the most important features for QSO classification in the ZTF data set. Using ZTFg-band data with at least 100 observational epochs per light curve, we obtain a 97% F1 score for QSOs. We find that with 3 day median cadence, a survey time span of at least 900 days is required to achieve a 90% QSO F1 score. However, one can obtain the same score with a survey time span of 1800 days and the median cadence prolonged to 12 days.