The widespread adoption of graph neural networks (GNNs) has brought increased attention to fairness issues related to sensitive attributes, such as gender and race, in practical scenarios. However, this concern remains largely unexplored in the context of graph clustering. Conventional fair graph clustering methods primarily depend on spectral clustering approaches. Meanwhile, we argue that existing graph learning works mainly focus on a single type of fairness, whereas graph clustering should achieve group equality-informed individual fairness. In this paper, we introduce for the first time a fairness-aware framework termed FairGC for deep graph clustering, which integrates the dual objectives of individual and group fairness while maintaining accurate clustering results. Specifically, we construct two views with distinct semantics using Siamese encoders. Then, we apply multi-step random walks on view-specific affinity graphs to capture high-order affinities of node pairs, thereby reformulating the contrastive learning with a focus on individual similarity. Besides, we utilize adversarial learning by making node representations independent of the estimated sensitive attributes to further eliminate group biases of clustering results. Extensive experiments on four benchmarks demonstrate the effectiveness and superiority of our proposed framework FairGC.