Hallucination detection in large language models (LLMs) requires balancing accu racy, efficiency, and robustness to distribution shift. Black-box consistency methods are effective but demand repeated inference; single-pass white-box probes are effi cient yet treat answer representations in isolation, often degrading sharply under domain shift. We propose QAOD (Question-Answer Orthogonal Decomposition), a single-pass framework that projects away the question-aligned direction from the answer representation to obtain a question-orthogonal component that suppresses domain-conditioned variation. To identify informative signals, QAOD further selects layers via diversity-penalized Fisher scoring and discriminative neurons via Fisher importance. To address both in-domain detection and cross-domain generalization, we design two complementary probing strategies: pairing the or thogonal component with question context yields a joint probe that maximizes in-domain discriminability, while using the orthogonal component alone preserves domain-agnostic factuality signals for robust transfer. QAOD's joint probe achieves the best in-domain AUROC across all evaluated model-dataset pairs, while the orthogonal-only probe delivers the strongest OOD transfer, surpassing the best white-box baseline by up to 21% on BioASQ at under 25% of generation cost.