Channel state information (CSI) reporting is important for multiple-input multiple-output (MIMO) transmitters to achieve high capacity and energy efficiency in frequency division duplex (FDD) mode. CSI reporting for massive MIMO systems could consume excessive bandwidth and degrade spectrum efficiency. Deep learning (DL)-based compression integrated with channel correlations have demonstrated success in improving CSI recovery. However, existing works focusing on CSI compression have shown little on the efficient encoding of CSI report. In this paper, we propose an efficient DL-based compression framework (called CQNet) to jointly tackle CSI compression, report encoding, and recovery under bandwidth constraint. CQNet can be directly integrated within other DL-based CSI feedback works for further enhancement. CQNet significantly outperforms solutions using uniform CSI quantization and $μ$-law non-uniform quantization. Compared with traditional CSI reporting, much fewer bits are required to achieve comparable CSI reconstruction accuracy.