Graph Neural Networks (GNNs) are powerful models capable of managing intricate connections in non-Euclidean data, such as social networks, physical systems, chemical structures, and communication networks. Despite their effectiveness, the large-scale and complex nature of graph data demands substantial computational resources and high performance during both training and inference stages, presenting significant challenges, particularly in the context of embedded systems. Recent studies on GNNs have investigated both software and hardware solutions to enhance computational efficiency. Earlier studies on Deep Neural Networks (DNNs) have indicated that methods like reconfigurable hardware and quantization are beneficial in addressing these issues. Unlike DNN research, studies on efficient computational methods for GNNs are less developed and require more exploration. This survey reviews the latest developments in quantization and FPGA-based acceleration for GNNs, showcasing the capabilities of reconfigurable systems (often FPGAs) to offer customized solutions in environments marked by significant sparsity and the necessity for dynamic load management. It also emphasizes the role of quantization in reducing both computational and memory demands through the use of fixed-point arithmetic and streamlined vector formats. This paper concentrates on low-power, resource-limited devices over general hardware accelerators and reviews research applicable to embedded systems. Additionally, it provides a detailed discussion of potential research gaps, foundational knowledge, obstacles, and prospective future directions.