Correlating Beijing–Arizona Sky Survey (BASS) data release 3 (DR3) catalogue with the ALLWISE data base, the data from optical and infrared information are obtained. The quasars from Sloan Digital Sky Survey are taken as training and test samples while those from LAMOST are considered as external test sample. We propose two schemes to onstruct the redshift estimation models with XGBoost, CatBoost, and Random Forest. One scheme (namely one-step model) is to predict photometric redshifts directly based on the optimal models created by these three algorithms; the other scheme (namely two-step model) is to first classify the data into low- and high-redshift data sets, and then predict photometric redshifts of these two data sets separately. For one-step model, the performance of these three algorithms on photometric redshift estimation is compared with different training samples, and CatBoost is superior to XGBoost and Random Forest. For two-step model, the performances of these three algorithms on the classification of low and high redshift subsamples are compared, and CatBoost still shows the best performance. Therefore, CatBoost is regarded as the core algorithm of classification and regression in two-step model. In contrast to one-step model, two-step model is optimal when predicting photometric redshift of quasars, especially for high-redshift quasars. Finally, the two models are applied to predict photometric redshifts of all quasar candidates of BASS DR3. The number of high-redshift quasar candidates is 3938 (redshift ≥3.5) and 121 (redshift ≥4.5) by two-step model. The predicted result will be helpful for quasar research and follow-up observation of high-redshift quasars.