Microbes have been demonstrated to be closely linked to diseases that pose a major threat to human health. Computing technologies can help researchers find potential microbe-drug associations more quickly and precisely. In this study, we introduced a novel computational prediction model called GLNNMDA based on global and local features of microbes and drugs to infer possible microbe-drug correlations. In GLNNMDA, we first constructed a heterogeneous network based on known microbe-drug relationships by integrating multiple similarity metrics of drugs and microbes. Subsequently, low-dimensional features will be extracted for nodes in the heterogeneous network by adopting the graph attention encoder. Next, based on combining these low-dimensional features with multiple properties of microbes and drugs to form a new comprehensive feature matrix, we would utilize the GLF module to extract the global and local features for microbes and drugs respectively, and then, we would further fuse these global and local features to come up with predictions of possible microbe-drug associations. Moreover, in order to evaluate the prediction performance of GLNNMDA, under the framework of fivefold cross-validation, intensive comparative experiments and case studies were done on different well-known public databases. The results showed that GLNNMDA obtained the highest AUC values as well as AUPR values of 0.9802 ± 0.0011, 0.9773 ± 0.0021 and 0.8586 ± 0.0004, 0.8008 ± 0.0031 in the two databases, MDAD and aBiofilm, respectively, compared to the state-of-the-art competing prediction methods. In addition, case studies of well-known microorganisms and drugs have demonstrated the effectiveness of GLNNMDA in inferring potential microbial drug associations, which implies that GLNNMDA may be a useful tool for microbe-drug association prediction in the future. The source code is available at: " https://github.com/KuangHaiYue/GLNNMDA.git ".