Model quantization is a compression technique that converts a full-precision model to a more compact low-precision version for better storage. Despite the great success of quantization, recent studies revealed the feasibility of malicious exploiting model quantization to implant quantization-conditioned backdoors (QCBs). These special backdoors remain dormant in full-precision models but are exposed upon quantization. Unfortunately, existing defenses have limited effects on mitigating QCBs. In this paper, we conduct the first in-depth analysis of QCBs. We reveal an intriguing characteristic of QCBs, where activation of backdoor-related neurons on even benign samples enjoy a distribution drift after quantization, although this drift is more significant on poisoned samples. Motivated by this finding, we propose to purify the backdoor-exposed quantized model by aligning its layer-wise activation with its full-precision version. To further exploit the more pronounced activation drifts on poisoned samples, we design an additional module to layer-wisely approximate poisoned activation distribution based on batch normalization statistics of the full-precision model. Extensive experiments are conducted, verifying the effectiveness of our defense and its resistance to the adaptive attack.