Deep Reinforcement Learning (DRL) has made significant advancements in various fields, such as autonomous driving, healthcare, and robotics, by enabling agents to learn optimal policies through interactions with their environments. However, the application of DRL in safety-critical domains presents challenges, particularly concerning the safety of the learned policies. DRL agents, which are focused on maximizing rewards, may select unsafe actions, leading to safety violations. Runtime safety monitoring is thus essential to ensure the safe operation of these agents, especially in unpredictable and dynamic environments. This paper introduces SMARLA, a black-box safety monitoring approach specifically designed for DRL agents. SMARLA utilizes machine learning to predict safety violations by observing the agent's behavior during execution. The approach is based on Q-values, which reflect the expected reward for taking actions in specific states. SMARLA employs state abstraction to reduce the complexity of the state space, enhancing the predictive capabilities of the monitoring model. Such abstraction enables the early detection of unsafe states, allowing for the implementation of corrective and preventive measures before incidents occur. We quantitatively and qualitatively validated SMARLA on three well-known case studies widely used in DRL research. Empirical results reveal that SMARLA is accurate at predicting safety violations, with a low false positive rate, and can predict violations at an early stage, approximately halfway through the execution of the agent, before violations occur. We also discuss different decision criteria, based on confidence intervals of the predicted violation probabilities, to trigger safety mechanisms aiming at a trade-off between early detection and low false positive rates.