Patients with diabetes are at increased risk of comorbid depression or anxiety, complicating their management. This study evaluated the performance of large language models (LLMs) in detecting these symptoms from secure patient messages. We applied multiple approaches, including engineered prompts, systemic persona, temperature adjustments, and zero-shot and few-shot learning, to identify the best-performing model and enhance performance. Three out of five LLMs demonstrated excellent performance (over 90% of F-1 and accuracy), with Llama 3.1 405B achieving 93% in both F-1 and accuracy using a zero-shot approach. While LLMs showed promise in binary classification and handling complex metrics like Patient Health Questionnaire-4, inconsistencies in challenging cases warrant further real-life assessment. The findings highlight the potential of LLMs to assist in timely screening and referrals, providing valuable empirical knowledge for real-world triage systems that could improve mental health care for patients with chronic diseases.