High-Efficiency Video Coding (HEVC) is the most recent video codec standard. It is substantial to analyze the HEVC steganography process due to its practical and academic significance. Thus, a secure HEVC steganography approach is introduced in this paper to study the possibility of hiding an encrypted secret audio message within a cover compressed video frame in a secure and complicated manner. In the preliminary stage, the secret audio message is compressed utilizing the Discrete Cosine Transform (DCT) to achieve a high capacity performance for the HEVC steganography process. After that, the suggested approach implies two-cascaded encryption layers to encrypt the compressed secret message before embedding it within a cover HEVC frame. In the first encryption layer, a novel encryption technique based on random projection and Legendre sequence in the Discrete Wavelet Transform (DWT) domain is introduced to cipher the compressed secret audio message. In the second encryption layer, the yielded encrypted audio message is represented in a form of quaternion numbers using the Quaternion Fast Fourier Transform (QFFT) technique. Each cover HEVC frame is also represented in a quaternion form. In the suggested approach, some straightforward quaternion mathematical operations are employed on the encrypted secret message and the cover HEVC frames to represent them in a quaternion form in the frequency domain, then the encrypted secret audio message is hidden within the cover HEVC frame. At the receiver, the secret message can be retrieved and extracted from the cover HEVC frame utilizing the same methodology of the employed quaternion mathematical operations. The major contributions of the suggested HEVC steganography scheme are: (1) it allows hiding of massive amount of secret information within cover video frames, and (2) it has higher robustness against multimedia attacks and steganalysis contrasted to the conventional and literature schemes. Furthermore, the proposed approach is evaluated utilizing different assessment metrics like Feature Similarity Index Measure (FSIM), Peak Signal-to-Noise Ratio (PSNR), correlation coefficient, and Structural Similarity Index Measure (SSIM) to evaluate the efficiency of the stego HEVC frames compared to the original ones. The achieved outcomes demonstrate that the suggested steganography scheme is straightforward to implement, more secure, and robust in the presence of steganalysis multimedia attacks compared to the literature approaches.