Channel reciprocity has been commonly assumed in time division duplex (TDD) massive multiple-input multiple-output (MIMO) communications, when acquiring downlink channel state information (CSI) at the base station through the uplink channel estimation. However, such channel reciprocity suffers from severe impairments when low-resolution analog-to-digital converters (ADCs) are introduced to reduce hardware cost and power consumption in practical systems. To compensate for such impairments, in this paper, we propose an efficient calibration state diagnosis scheme built upon compressive sensing techniques, leveraging the sparsity property of the calibration operations at the BS antennas under an additive quantization noise model. Compared with the traditional pilot-based approaches, our proposed scheme achieves comparable accuracy performance with 1-2 quantization bits and hence significantly reduces pilot overhead therein.