Uncertainty quantification (UQ) in foundation models is crucial for identifying and mitigating hallucinations in automatically generated text. However, heuristic UQ approaches lack statistical guarantees for key metrics such as the false discovery rate (FDR) in selective prediction tasks. Previous research adopts the split conformal prediction (SCP) framework to ensure desired coverage of admissible answers by constructing data-driven prediction sets, yet these sets typically contain incorrect candidates, undermining their practical effectiveness. To address this, we introduce COIN, an uncertainty-guarding selection framework that calibrates statistically valid uncertainty thresholds to filter a single generated answer per question under user-specified FDR constraints. COIN estimates the empirical error rate on the calibration set and applies confidence interval methods such as Clopper–Pearson to establish a high-probability upper bound on the true error rate (i.e., FDR). This enables the selection of the largest threshold that ensures FDR control on test data while significantly increasing sample retention. We demonstrate COIN's robustness in risk control, strong test-time power in retaining admissible answers, and predictive efficiency under limited calibration data across both general and multimodal text generation tasks. Furthermore, we show that employing alternative UQ and upper bound construction strategies can further boost COIN's power performance, which underscores its extensibility and adaptability to diverse application scenarios.