Major depressive disorder (MDD) is the most common psychological disease. To improve the recognition accuracy of MDD, more and more machine learning methods have been proposed to mining EEG features, i.e. typical brain functional patterns and recognition methods that are closely related to depression using resting EEG signals. Most existing methods typically utilize threshold methods to filter weak connections in the brain functional connectivity network (BFCN) and construct quantitative statistical features of brain function to measure the BFCN.However, these thresholds may excessively remove weak connections with functional relevance, which is not conducive to discovering potential hidden patterns in weak connections. In addition, statistical features cannot describe the topological structure features and information network propagation patterns of the brain's different functional regions. To solve these problems, we propose a novel MDD recognition method based on a multi-granularity graph convolution network (MGGCN). On the one hand, this method applies multiple sets of different thresholds to build a multi-granularity functional neural network, which can remove noise while fully retaining valuable weak connections. On the other hand, this method utilizes graph neural network to learn the topological structure features and brain saliency patterns of changes between brain functional regions on the multi-granularity functional neural network. Experimental results on the benchmark datasets validate the superior performance and time complexity of MGGCN. The analysis shows that as the granularity increases, the connectivity defects in the right frontal(RF) and right temporal (RT) regions, left temporal(LT) and left posterior(LP) regions increase. The brain functional connections in these regions can serve as potential biomarkers for MDD recognition.