While high-accuracy machine learning (ML) models for predicting student learning performance have been widely explored, their deployment in real educational settings can lead to unintended harm if the predictions are biased. This study systematically examines the trade-offs between prediction accuracy and fairness in ML models trained on the widely used Open University Learning Analytics Dataset (OULAD). We evaluated the relationship between model accuracy and fairness across various student demographic subgroups and investigated the extent to which fairness can be improved without significantly sacrificing accuracy. Our analysis revealed that standard ML models often exhibit bias; however, applying bias mitigation techniques can reduce these disparities while maintaining acceptable accuracy. Our findings emphasize the importance of auditing ML models for fairness to ensure that predictive insights are equitable across diverse student populations. We also discuss implications for best practices and challenges in achieving fair ML models for student performance prediction.