Large Language Models (LLMs) have shown significant potential as judges for Machine Translation (MT) quality assessment, providing both scores and fine-grained feedback. Although approaches such as GEMBA-MQM have shown state-of-the-art performance on reference-free evaluation, the predicted errors do not align well with those annotated by human, limiting their interpretability as feedback signals. To enhance the quality of error annotations predicted by LLM evaluators, we introduce a universal and training-free framework, $\textbf{MQM-APE}$, based on the idea of filtering out non-impactful errors by Automatically Post-Editing (APE) the original translation based on each error, leaving only those errors that contribute to quality improvement. Specifically, we prompt the LLM to act as 1) $\textit{evaluator}$ to provide error annotations, 2) $\textit{post-editor}$ to determine whether errors impact quality improvement and 3) $\textit{pairwise quality verifier}$ as the error filter. Experiments show that our approach consistently improves both the reliability and quality of error spans against GEMBA-MQM, across eight LLMs in both high- and low-resource languages. Orthogonal to trained approaches, MQM-APE complements translation-specific evaluators such as Tower, highlighting its broad applicability. Further analysis confirms the effectiveness of each module and offers valuable insights into evaluator design and LLMs selection.