In this paper, we consider the problem of underdetermined direction-of-arrival (DOA) estimation using coprime arrays from a ont-bit perspective, where the coarray correlations of the quantized sparse measurements are explored for augmented covariance matrix reconstruction. To fully utilize the coarray signals calculated from the one-bit coprime array measurements for DOA estimation, a correlation reconstruction problem is formulated to obtain the quantized covariance matrix corresponding to a filled coarray containing the discontiguous one, where the one-bit quantization transforms the possibilities of correlations from an infinite to a finite number. The performance of the proposed method is validated from the aspects of degrees-of-freedom (DOFs), estimation accuracy, as well as the resolution performance. Simulation results demonstrate that the proposed method not only retains full achievable DOFs of the coprime array, but is also capable of presenting a better DOA estimation performance than the non-quantization approaches.