Large language models (LLMs) have shown impressive capabilities in tasks such as machine translation, text summarization, question answering, and solving complex mathematical problems. However, their primary training on data-rich languages like English limits their performance in low-resource languages. This study addresses this gap by focusing on the Indexical Shift problem in Turkish. The Indexical Shift problem involves resolving pronouns in indexical shift contexts, a grammatical challenge not present in high-resource languages like English. We present the first study examining indexical shift in any language, releasing a Turkish dataset specifically designed for this purpose. Our Indexical Shift Dataset consists of 156 multiple-choice questions, each annotated with necessary linguistic details, to evaluate LLMs in a few-shot setting. We evaluate recent multilingual LLMs, including GPT-4, GPT-3.5, Cohere-AYA, Trendyol-LLM, and Turkcell-LLM, using this dataset. Our analysis reveals that even advanced models like GPT-4 struggle with the grammatical nuances of indexical shift in Turkish, achieving only moderate performance. These findings underscore the need for focused research on the grammatical challenges posed by low-resource languages. We released the dataset and code here.