Compressed sensing shows that a sparse signal can stably be recovered from incomplete linear measurements. But, in practical applications, some signals have additional structure, where the nonzero elements arise in some blocks. We call such signals as block-sparse signals. In this paper, the $\ell_2/\ell_1-\alpha\ell_2$ minimization method for the stable recovery of block-sparse signals is investigated. Sufficient conditions based on block mutual coherence property and associating upper bound estimations of error are established to ensure that block-sparse signals can be stably recovered in the presence of noise via the $\ell_2/\ell_1-\alpha\ell_2$ minimization method. For all we know, it is the first block mutual coherence property condition of stably reconstructing block-sparse signals by the $\ell_2/\ell_1-\alpha\ell_2$ minimization method. Additionally, the numerical experiments implemented verify the performance of the $\ell_2/\ell_1-\alpha\ell_2$ minimization.