With the high efficiency and commendable performance, quaternion orthogonal matching pursuit (QOMP) has gained significant attention in the field of sparse signal processing. However, QOMP struggles to leverage block structure information in data, which can result in suboptimal outcomes. In this paper, we propose a quaternion block orthogonal matching pursuit (QBOMP) model to effectively incorporate the block prior information for the hypercomplex quaternion signals. To address the challenges posed by the non-commutative multiplication of quaternion, we present a quaternion optimization algorithm of QBOMP. Additionally, we establish a necessary condition for QBOMP to reliably recover quaternion sparse signals through theoretical analysis. Meanwhile, since real-world data is often affected by noise, we propose a robust quaternion block orthogonal matching pursuit method, termed RQBOMP. It employs fair function in M-estimation as loss function instead of the quaternion mean square error (QMSE) and can be successfully optimized by half-quadratic theory and Alternating Direction Method of Multipliers (ADMM) framework. Experimental results on both synthetic and real-world data demonstrate that RQBOMP has robustness and superiority in quaternion signals recovery, color image recovery and color face recognition tasks.