Machine translation is essential for international communication and is extensively used across numerous human-related applications. Despite substantial progress, fairness issues persist in current machine translation systems. This paper addresses the intersection of machine translation testing and fairness testing, two critical and widely explored domains in software engineering. We introduce FairMT, an automated fairness testing approach specifically designed for evaluating machine translation systems. FairMT operates on the assumption that translations of semantically similar sentences, containing protected attributes from distinct demographic groups, should maintain comparable meanings. It comprises three key steps: (1) test input generation, producing inputs covering various demographic groups based on metamorphic relations; (2) test oracle generation, identifying potential unfair translations based on semantic similarity measurements; and (3) regression, discerning genuine fairness issues from those caused by low-quality translation. Leveraging FairMT, we conduct an empirical study on three leading machine translation systems—Google Translate, T5, and Transformer. Our investigation uncovers up to 832, 1,984, and 2,627 unfair translations across the three systems, respectively. Intriguingly, we observe that fair translations tend to exhibit better translation performance, challenging the conventional wisdom of a fairness-performance trade-off prevalent in the fairness literature.