Uncertainty quantification is a set of techniques that measure confidence in language models. They can be used, for example, to detect hallucinations or alert users to review uncertain predictions. To be useful, these confidence scores must be correlated with the quality of the output. However, recent work found that fine-tuning can affect the correlation between confidence scores and quality. Hence, we investigate the underlying behavior of confidence scores to understand its sensitivity to supervised fine-tuning (SFT). We find that post-SFT, the correlation of various confidence scores degrades, which can stem from changes in confidence scores due to factors other than the output quality, such as the output's similarity to the training distribution. We demonstrate via a case study how failing to address this miscorrelation reduces the usefulness of the confidence scores on a downstream task. Our findings show how confidence metrics cannot be used off-the-shelf without testing, and motivate the need for developing metrics which are more robust to fine-tuning.