Quantization is widely used to accelerate inference and streamline the deployment of large language models (LLMs), yet its effects on self-explanations (SEs) remain unexplored. SEs, generated by LLMs to justify their own outputs, require reasoning about the model's own decision-making process, a capability that may exhibit particular sensitivity to quantization. As SEs are increasingly relied upon for transparency in high-stakes applications, understanding whether and to what extent quantization degrades SE quality and faithfulness is critical. To address this gap, we examine two types of SEs: natural language explanations (NLEs) and counterfactual examples, generated by LLMs quantized using three common techniques at distinct bit widths. Our findings indicate that quantization typically leads to moderate declines in both SE quality (up to 4.4\%) and faithfulness (up to 2.38\%). The user study further demonstrates that quantization diminishes both the coherence and trustworthiness of SEs (up to 8.5\%). Compared to smaller models, larger models show limited resilience to quantization in terms of SE quality but better maintain faithfulness. Moreover, no quantization technique consistently excels across task accuracy, SE quality, and faithfulness. Given that quantization's impact varies by context, we recommend validating SE quality for specific use cases, especially for NLEs, which show greater sensitivity. Nonetheless, the relatively minor deterioration in SE quality and faithfulness does not undermine quantization's effectiveness as a model compression technique.