This paper considers the trajectory design problem for unmanned aerial vehicles (UAVs) via meta-reinforcement learning. It is assumed that the UAV can move in different directions to explore a specific area and collect data from the ground nodes (GNs) located in the area. The goal of the UAV is to reach the destination and maximize the total data collected during the flight on the trajectory while avoiding collisions with other UAVs. In the literature on UAV trajectory designs, vanilla learning algorithms are typically used to train a task-specific model, and provide near-optimal solutions for a specific spatial distribution of the GNs. However, this approach requires retraining from scratch when the locations of the GNs vary. In this work, we propose a meta reinforcement learning framework that incorporates the method of Model-Agnostic Meta-Learning (MAML). Instead of training task-specific models, we train a common initialization for different distributions of GNs and different channel conditions. From the initialization, only a few gradient descents are required for adapting to different tasks with different GN distributions and channel conditions. Additionally, we also explore when the proposed MAML framework is preferred and can outperform the compared algorithms.