Early detection of cognitive impairment is critical for timely intervention; yet, traditional screening instruments such as the Mini-Mental State Examination (MMSE) require trained personnel and face-to-face administration, limiting their scalability. This work explores whether a large language model (LLM) can be adapted to detect cognitive impairment from conversational transcripts in a low-resource setting. We formulate the task as a binary classification problem (normal vs.impaired) using transcripts from the ADReSS dataset, in which patients describe the Cookie Theft picture. A LLaMA 3.2-1B-Instruct model is adapted to this task through Quantized Low-Rank Adaptation (QLoRA), which keeps the base model frozen and quantized while training only a small set of adapter parameters (approximately 1.8% of the total). To quantify the effect of adaptation, we compare the fine-tuned model against the same architecture without any task-specific training. The frozen baseline collapses to majority-class prediction (accuracy 0.529, macro- \(F_1\) 0.346), confirming the absence of discriminative ability. In contrast, the QLoRA-adapted model achieves 0.941 accuracy and 0.940 macro- \(F_1\) on a stratified validation split, correctly classifying both classes with high confidence. These results demonstrate that parameter-efficient fine-tuning can yield effective cognitive screening models from conversational language even under limited data and compute constraints.