In recent years, fairness in machine learning has emerged as a critical concern to guarantee that developed and deployed predictive models do not have distinct predictions for marginalized groups. However, it is essential to avoid biased decisions and promote equitable outcomes dealing with multiple (sub)group attributes (gender, race, etc.) simultaneously. In this work, we consider applying subgroup justice concepts to gradient-boosting machines designed for supervised learning problems. Our approach expanded gradient-boosting methodologies to explore a broader range of objective functions, which combines conventional losses such as the ones from classification and regression and a min-max fairness term. The optimization process explored the primal-dual problems at each boosting round. This generic framework can be adapted to diverse fairness concepts. The proposed min-max primal-dual gradient boosting algorithm was empirically shown to be a powerful and flexible approach to address binary and subgroup fairness.