To address the challenges posed by the deployment of microservices of future end-user applications in the cloud continuum, a performance prediction model working together with a network elasticity controller will be needed. With that aim, this work introduces Layered Queuing-Graph Neural Networks (LQ-GNN), a novel Machine earning (ML) approach to develop a generalized performance prediction model for microservicebased plications. Unlike previous works focused on individual applications, our proposal aims for a versatile model applicable to any microservice-based application, integrating the Layered Queueing Network (LQN) modeling with Graph Neural Networks (GNN). LQ-GNN allows to efficiently estimate the response time of applications under different resource allocations and placements on the computing continuum. The obtained evaluation results indicate that the roposed model achieves a prediction error below 10% when considering different evaluation scenarios. Compared to existing methodologies, our approach balances prediction accuracy and computational efficiency, making it viable for real-time deployments. Consequently, ML-based performance prediction can significantly enhance the resource management and elasticity control of microservice-based architectures, leading to more resilient and efficient systems.