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The development of fair machine learning models that effectively avert bias and discrimination is an important problem that has garnered attention in recent years. The necessity of encoding complex relational dependencies among the features and variables for competent predictions require the development of fair, yet expressive relational models. In this work, we introduce Fair-A3SL, a fairness-aware structure learning algorithm for learning relational structures, which incorporates fairness measures while learning relational graphical model structures. Our approach is versatile in being able to encode a wide range of fairness metrics such as statistical parity difference, overestimation, equalized odds, and equal opportunity, including recently proposed relational fairness measures. While existing approaches employ the fairness measures on pre-determined model structures post prediction, Fair-A3SL directly learns the structure while optimizing for the fairness measures and hence is able to remove any structural bias in the model. We demonstrate the effectiveness of our learned model structures when compared with the state-of-the-art fairness models quantitatively and qualitatively on datasets representing three different modeling scenarios: i) a relational dataset, ii) a recidivism prediction dataset widely used in studying discrimination, and iii) a recommender systems dataset. Our results show that Fair-A3SL can learn fair, yet interpretable and expressive structures capable of making accurate predictions.