Passive acoustic monitoring (PAM) techniques have shown great potential in studying underwater gas plumes by leveraging bubble resonance signals. Traditional bubble detection methods generally operate on the mixture of bubble and ambient sounds, which cannot achieve satisfactory performance in low SNR environments. In this study, we propose a deep learning (DL) based bubble sound separation method to extract the bubble waveform from the noisy mixture prior to detection, thereby enhancing the bubble detection performance. To obtain the labeled training data, we developed a numerical simulation framework based on bubble acoustic theories to generate the ground truth bubble sounds, which are then mixed with diverse noises. Experiments were conducted with both simulated data and realistic PAM recordings. The simulation experiments under different noise conditions demonstrate that the DL models can effectively extract bubble sound, even when their features are barely visible in the time-frequency domain. In the real-world experiment, the trained model was applied to the PAM recordings collected in Haima cold seep, and we found a negative correlation between bubble release rate and ambient pressure when the hydrophones were near the gas plumes, which is in accordance with existing literature and further validates the proposed method's effectiveness.