Large language models (LLMs) exhibit biases against LGBTQ+ populations. While safety training may reduce explicit bias, implicit stereotype-driven associations often persist. In this work, we examine implicit bias toward transgender people in two main scenarios. First, we adapt word association tests to measure whether LLMs disproportionately pair negative concepts with "transgender" and positive concepts with "cisgender". Second, given the systemic challenges transgender people face in healthcare, we design a healthcare appointment allocation task where models act as scheduling agents choosing between cisgender and transgender candidates across stereotype-prone specialties. We evaluate seven LLMs in English and Spanish. Our results show consistent bias in categories such as appearance, risk, and veracity, indicating stronger negative associations with transgender individuals. In the allocation task, transgender candidates are favored for STI and mental health services, while cisgender candidates are preferred in gynecology and breast care. Notably, this bias is reduced when patient symptoms accompany demographic information. These findings underscore the need for research that address subtle stereotype-driven biases in LLMs to ensure equitable treatment of transgender people in healthcare applications.