Face Recognition (FR) plays a crucial role in many critical (high-stakes) applications, where errors in the recognition process can lead to serious consequences. Face Image Quality Assessment (FIQA) techniques enhance FR systems by providing quality estimates of face samples, enabling the systems to discard samples that are unsuitable for reliable recognition or lead to low-confidence recognition decisions. Most state-of-the-art FIQA techniques rely on extensive supervised training to achieve accurate quality estimation. In contrast, unsupervised techniques eliminate the need for additional training but tend to be slower and typically exhibit lower performance. In this paper, we introduce FROQ (Face Recognition Observer of Quality), a semi-supervised, training-free approach that leverages specific intermediate representations within a given FR model to estimate face-image quality, and combines the efficiency of supervised FIQA models with the training-free approach of unsupervised methods. A simple calibration step based on pseudo-quality labels allows FROQ to uncover specific representations, useful for quality assessment, in any modern FR model. To generate these pseudo-labels, we propose a novel unsupervised FIQA technique based on sample perturbations. Comprehensive experiments with four state-of-the-art FR models and eight benchmark datasets show that FROQ leads to highly competitive results compared to the state-of-the-art, achieving both strong performance and efficient runtime, without requiring explicit training.