Large language models (LLMs) are reported to be susceptible to reliability issues including hallucinations and reasoning errors. Existing quality detectors can predict unreliable outputs based on internal representations with decent accuracies. However, training these detectors often requires substantial domain-specific labeled data, which limits their applicability to new domains. We propose to address this challenge by transferring representational signatures across related domains. More specifically, we build a statistical framework that first extracts low-dimensional signatures for each domain, then uses cross-fitting to learn composition weights of domain-specific predictors. The weights adapt to each source domain's predictive relevance to the target domain. Our asymptotic analysis of a least-squares analogue characterizes the limiting prediction risk and aggregation weights in terms of domain discrepancies and representation bias. Experiments across multiple hallucination datasets and open-weight LLMs show that, with only 25 labeled questions in the new domain, adaptive transfer improves hallucination-detection AUROC by 6.8%. For reasoning-step prediction, it improves AUROC by 8.4%.