Currently, short video platforms have become the primary place for individuals to share experiences and obtain information. To better meet users' needs for acquiring information while browsing short videos, some apps have introduced a search entry at the bottom of videos, accompanied with recommended relevant queries. This scenario is known as query recommendation in video-related search, where core task is item-to-query (I2Q) recommendation. As this scenario has only emerged in recent years, there is a notable scarcity of academic research and publicly available datasets in this domain. To address this gap, we systematically examine the challenges associated with this scenario for the first time. Subsequently, we release a large-scale dataset derived from real-world data pertaining to the query recommendation in video-\textit{\textbf{r}}elated \textit{\textbf{s}}earch on the \textit{\textbf{Kuai}}shou app (\textbf{KuaiRS}). Presently, existing methods rely on embeddings to calculate similarity for matching short videos with queries, lacking deep interaction between the semantic content and the query. In this paper, we introduce a novel LLM-based framework named \textbf{GREAT}, which \textit{\textbf{g}}uides que\textit{\textbf{r}}y g\textit{\textbf{e}}ner\textit{\textbf{a}}tion with a \textit{\textbf{t}}rie to address I2Q recommendation in related search. Specifically, we initially gather high-quality queries with high exposure and click-through rate to construct a query-based trie. During training, we enhance the LLM's capability to generate high-quality queries using the query-based trie. In the inference phase, the query-based trie serves as a guide for the token generation. Finally, we further refine the relevance and literal quality between items and queries via a post-processing module. Extensive offline and online experiments demonstrate the effectiveness of our proposed method.