This paper introduces Query Harmfulness Prediction (QHP), a novel extension of Query Performance Prediction that focuses on predicting the potential harmfulness of search results. While traditional QPP methods predict standard retrieval metrics, QHP addresses the growing need for safer information retrieval by anticipating when queries might return harmful but topically relevant results. We investigate three families of predictors: classical pre-retrieval QPP methods, LLM-based strategies leveraging signals such as controversy and misinformation, and a query quality classifier adapted from prior work. Using datasets from TREC and CLEF campaigns, we evaluate these approaches with compatibility harmful as the target measure. Our results show that while traditional QPP predictors capture limited signals of harmfulness, LLM-based methods consistently provide stronger correlations, especially on high-risk queries. These findings establish QHP as a timely research direction for developing safer retrieval systems that balance relevance with user safety.