We study how multiple notions of fairness can be incorporated into a single Bayesian non-parametric regression framework for insurance pricing, with a focus on claim frequency modeling under a log-link. We consider a Generalized Gaussian Process Regression (GGPR) model for count data with risk exposure and introduce fairness interventions in its architecture. Specifically, we address notions of individual fairness by altering the kernel structure to control the similarity between policies. We also address group-level fairness by enforcing demographic parity through constraints affecting the posterior. This modified GGPR architecture allows us to jointly enforce multiple fairness notions within a single probabilistic model. We empirically explore trade-offs with portfolio balance to prevent systematic overpricing or underpricing at the portfolio level, and how different fairness criteria interact when combined. The results highlight the importance of adopting a multi-criteria, context-aware approach to fairness in insurance pricing.