Translation Quality Estimation (TQE) is an essential step before deploying the output translation into usage. TQE is also critical in assessing machine translation (MT) and human translation (HT) quality without seeing the reference translations. This work examines whether the state-of-the-art large language models (LLMs) can be fine-tuned for the TQE task and their capability. We take ChatGPT as one example and approach TQE as a binary classification task. Using \textbf{eight language pairs} including English to Italian, German, French, Japanese, Dutch, Portuguese, Turkish, and Chinese training corpora, our experimental results show that fine-tuned ChatGPT via its API can achieve a relatively high score on predicting translation quality, i.e. \textit{if the translation needs to be edited}. However, there is definitely much space to improve the model accuracy, e.g. they are 82.42\% and 83.69\% for English-Italian and English-German respectively using our experimental settings. English-Italiano bilingual Abstract is available in the paper.