Machine-learning-based models have recently gained traction as a way to overcome the slow downstream implementation process of FPGAs by building models that provide fast and accurate performance predictions. However, these models suffer from two main limitations: (1) training requires large amounts of data (features extracted from FPGA synthesis and implementation reports), which is cost-inefficient because of the time-consuming FPGA design cycle; (2) a model trained for a specific environment cannot predict for a new, unknown environment. In a cloud system, where getting access to platforms is typically costly, data collection for ML models can significantly increase the total cost-ownership (TCO) of a system. To overcome these limitations, we propose LEAPER, a transfer learning-based approach for FPGA-based systems that adapts an existing ML-based model to a new, unknown environment to provide fast and accurate performance and resource utilization predictions. Experimental results show that our approach delivers, on average, 85% accuracy when we use our transferred model for prediction in a cloud environment with 5-shot learning and reduces design-space exploration time by 10x, from days to only a few hours.