Pretrained model-based evaluation metrics have demonstrated strong performance with high correlations with human judgments in various natural language generation tasks such as image captioning. Despite the impressive results, their impact on fairness is under-explored -- it is widely acknowledged that pretrained models can encode societal biases, and utilizing them for evaluation purposes may inadvertently manifest and potentially amplify biases. In this paper, we conduct a systematic study in gender biases of model-based evaluation metrics with a focus on image captioning tasks. Specifically, we first identify and quantify gender biases in different evaluation metrics regarding profession, activity, and object concepts. Then, we demonstrate the negative consequences of using these biased metrics, such as favoring biased generation models in deployment and propagating the biases to generation models through reinforcement learning. We also present a simple but effective alternative to reduce gender biases by combining n-gram matching-based and pretrained model-based evaluation metrics.