This paper studies the problem of information retrieval, to adapt to unseen tasks. Existing work generates synthetic queries from domain-specific documents to jointly train the retriever. However, the conventional query generator assumes the query as a question, thus failing to accommodate general search intents. A more lenient approach incorporates task-adaptive elements, such as few-shot learning with an 137B LLM. In this paper, we challenge a trend equating query and question, and instead conceptualize query generation task as a "compilation" of high-level intent into task-adaptive query. Specifically, we propose EGG, a query generator that better adapts to wide search intents expressed in the BeIR benchmark. Our method outperforms baselines and existing models on four tasks with underexplored intents, while utilizing a query generator 47 times smaller than the previous state-of-the-art. Our findings reveal that instructing the LM with explicit search intent is a key aspect of modeling an effective query generator.