This paper presents a quaternion block sparse representation (QBSR) method for structural sparse signal recovery in quaternion space. Due to the noncommutativity of quaternion multiplication, conventional optimization algorithms originally designed for real-valued and complex-valued signal recovery problems are not applicable to the optimization problem of QBSR. To combat this problem, we leverage several quaternion operators and devise an effective algorithm for QBSR within the ADMM (Alternating Direction Method of Multipliers) framework. The second contribution of this work is to develop a QBSR based classifier referred to as QBSRC for quaternion data classification with application to color face recognition. Compared with real-valued representation based classifiers handling multiple color channels of a color image independently, QBSRC treats a color image as a quaternion signal and represents it in a holistic manner. The third contribution is to provide the theoretical analysis of QBSRC and rigorously prove that QBSRC is guaranteed to succeed in classification of any new test sample under appropriate condition. Experiments on both synthetic and real-world datasets demonstrate the effectiveness of the proposed method for quaternion signal recovery and classification.