Predicting the number of infections in the anti-epidemic process is extremely beneficial to the government in developing anti-epidemic strategies, especially in fine-grained geographic units. Previous works focus on low spatial resolution prediction, e.g., county-level, and preprocess data to the same geographic level, which loses some useful information. In this paper, we propose a fine-grained population mobility data-based model (FGC-COVID) utilizing data of two geographic levels for community-level COVID-19 prediction. We use the population mobility data between Census Block Groups (CBGs), which is a finer-grained geographic level than community, to build the graph and capture the dependencies between CBGs using graph neural networks (GNNs). To mine as finer-grained patterns as possible for prediction, a spatial weighted aggregation module is introduced to aggregate the embeddings of CBGs to community level based on their geographic affiliation and spatial autocorrelation. Extensive experiments on 300 days LA city COVID-19 data indicate our model outperforms existing forecasting models on community-level COVID-19 prediction.