Illicit drug use among teenagers and young adults (TYAs) remains a pressing public health concern, with rising prevalence and long-term impacts on health and well-being. To detect illicit drug use among TYAs, researchers analyze large-scale surveys such as the Youth Risk Behavior Survey (YRBS) and the National Survey on Drug Use and Health (NSDUH), which preserve rich demographic, psychological, and environmental factors related to substance use. However, existing modeling methods treat survey variables independently, overlooking latent and interconnected structures among them. To address this limitation, we propose LAMI (LAtent relation Mining with bi-modal Interpretability), a novel joint graph-language modeling framework for detecting illicit drug use and interpreting behavioral risk factors among TYAs. LAMI represents individual responses as relational graphs, learns latent connections through a specialized graph structure learning layer, and integrates a large language model to generate natural language explanations grounded in both graph structures and survey semantics. Experiments on the YRBS and NSDUH datasets show that LAMI outperforms competitive baselines in predictive accuracy. Interpretability analyses further demonstrate that LAMI reveals meaningful behavioral substructures and psychosocial pathways, such as family dynamics, peer influence, and school-related distress, that align with established risk factors for substance use.