Adaptive coding in quantum communication offers a promising approach to enhance the efficiency and reliability of data transmission using quantum superposition, particularly in noisy and error-prone channels. This study investigates the effectiveness of quantum communication combined with adaptive coding for compressed image transmission using Joint Photographic Experts Group (JPEG) codec and High Efficiency Image Format (HEIF) with polar codes at varying rates. To maintain a similar bandwidth, the source coding rates are adjusted according to the channel coding rates. The results show that the adaptive coding based quantum communication system significantly outperforms equivalent classical systems, especially at low signal-to-noise ratios (SNR), achieving Peak Signal-to-Noise Ratio (PSNR) improvements up to 65 dB and Structural Similarity Index Measure (SSIM) values up to 0.9999 for HEIF images and PSNR values up to 58 dB with SSIM values up to 0.9994 for JPEG images. These findings demonstrate the superior robustness and higher image quality of adaptive coding based quantum communication in varying channel noise conditions and bandwidth restricted applications.