As the utilization of language models in interdisciplinary, human-centered studies grow, expectations of their capabilities continue to evolve. Beyond excelling at conventional tasks, models are now expected to perform well on user-centric measurements involving confidence and human (dis)agreement- factors that reflect subjective preferences. While modeling subjectivity plays an essential role in cognitive science and has been extensively studied, its investigation at the intersection with NLP remains under-explored. In light of this gap, we explore how language models can quantify subjectivity in cognitive appraisal by conducting comprehensive experiments and analyses with both fine-tuned models and prompt-based large language models (LLMs). Our quantitative and qualitative results demonstrate that personality traits and demographic information are critical for measuring subjectivity, yet existing post-hoc calibration methods often fail to achieve satisfactory performance. Furthermore, our in-depth analysis provides valuable insights to guide future research at the intersection of NLP and cognitive science.