Large language models (LLMs) often struggle in specialized domains such as legal reasoning due to limited expert knowledge, resulting in factually incorrect outputs or hallucinations. This paper presents an effective method for adapting advanced LLMs to German legal question answering through a novel synthetic data generation approach. In contrast to costly human-annotated resources or unreliable synthetic alternatives, our approach systematically produces high-quality, diverse, and legally accurate question-answer pairs directly from authoritative German statutes. Using rigorous automated filtering methods and parameter-efficient fine-tuning techniques, we demonstrate that LLMs adapted with our synthetic dataset significantly outperform their baseline counterparts on German legal question answering tasks. Our results highlight the feasibility of using carefully designed synthetic data as a robust alternative to manual annotation in high-stakes, knowledge-intensive domains.