Adaptive immune system uses T cell receptors (TCRs) to recognize pathogens and to consequently initiate immune responses. TCRs can be sequenced from individuals and methods analyzing the specificity of the TCRs can help us better understand individuals' immune status in different disorders. For this task, we have developed TCRGP, a novel Gaussian process method that predicts if TCRs recognize specified epitopes. TCRGP can utilize the amino acid sequences of the complementarity determining regions (CDRs) from TCRα and TCRβ chains and learn which CDRs are important in recognizing different epitopes. Our comprehensive evaluation with epitope-specific TCR sequencing data shows that TCRGP achieves on average higher prediction accuracy in terms of AUROC score than existing state-of-the-art methods in epitope-specificity predictions. We also propose a novel analysis approach for combined single-cell RNA and TCRαβ (scRNA+TCRαβ) sequencing data by quantifying epitope-specific TCRs with TCRGP and identify HBV-epitope specific T cells and their transcriptomic states in hepatocellular carcinoma patients.