People increasingly rely on online sources for health information seeking due to their convenience and timeliness, traditionally using search engines like Google as the primary search agent. Recently, the emergence of generative Artificial Intelligence (AI) has made Large Language Model (LLM) powered conversational agents such as ChatGPT a viable alternative for health information search. However, while trust is crucial for adopting the online health advice, the factors influencing people's trust judgments in health information provided by LLM-powered conversational agents remain unclear. To address this, we conducted a mixed-methods, within-subjects lab study (N=21) to explore how interactions with different agents (ChatGPT vs. Google) across three health search tasks influence participants' trust judgments of the search results as well as the search agents themselves. Our key findings showed that: (a) participants' trust levels in ChatGPT were significantly higher than Google in the context of health information seeking; (b) there is a significant correlation between trust in health-related information and trust in the search agent, however only for Google; (c) the type of search tasks did not affect participants' perceived trust; and (d) participants' prior knowledge, the style of information presentation, and the interactive manner of using search agents were key determinants of trust in the health-related information. Our study taps into differences in trust perceptions when using traditional search engines compared to LLM-powered conversational agents. We highlight the potential role LLMs play in health-related information-seeking contexts, where they excel as stepping stones for further search. We contribute key factors and considerations for ensuring effective and reliable personal health information seeking in the age of generative AI.