Fibromyalgia syndrome (FMS) is a type of rheumatology that seriously affects the normal life of patients. Due to the complex clinical manifestations of FMS, it is challenging to detect FMS. Therefore, an automatic FMS diagnosis model is urgently needed to assist physicians. Brain functional connectivity networks (BFCNs) constructed by resting-state functional magnetic resonance imaging (rs-fMRI) to describe brain functions have been widely used to identify individuals with relevant diseases from normal control (NC). Therefore, we propose a novel model based on BFCN and graph convolutional network (GCN) for automatic FMS diagnosis. Firstly, a novel fused BFCN method is proposed by fusing Pearson's correlation (PC) and low-rank (LR) BFCN, which retains information and reduces data redundancy to construct BFCN. Then we combine the feature of BFCN with non-image information of subjects to obtain nodes and adjacency matrices, which builds a graph with edge attention. Finally, the graph is sent to the GCN layer for FMS diagnosis. Our model is evaluated on the in-house FMS dataset to achieve 82.48% accuracy. The experimental results show that our method outperforms the state-of-the-art competing methods.