Exploring the functions of genes and gene products is crucial to a wide range of fields, including medical research, evolutionary biology, and environmental science. However, discovering new functions largely relies on expensive and exhaustive wet lab experiments. Existing methods of automatic function annotation or prediction mainly focus on protein function prediction with sequence, 3D-structures or protein family information. In this study, we propose to tackle the gene function prediction problem by exploring Gene Ontology graph and annotation with BERT (GoBERT) to decipher the underlying relationships among gene functions. Our proposed novel function prediction task utilizes existing functions as inputs and generalizes the function prediction to gene and gene products. Specifically, two pre-train tasks are designed to jointly train GoBERT to capture both explicit and implicit relations of functions. Neighborhood prediction is a self-supervised multi-label classification task that captures the explicit function relations. Specified masking and recovering task helps GoBERT in finding implicit patterns among functions. The pre-trained GoBERT possess the ability to predict novel functions for various gene and gene products based on known functional annotations. Extensive experiments, biological case studies, and ablation studies are conducted to demonstrate the superiority of our proposed GoBERT.