Precise and reliable measurement of wind power uncertainty plays a significant role in the economic operation and real-time control of the smart grid. In this paper, a novel spiking neural network (SNN) architecture is proposed for solving regression tasks, and a multi-objective gradient descent (MOGD) algorithm is employed to generate high-quality wind power prediction intervals (PIs). SNNs improve upon conventional artificial neural networks (ANNs) by encoding interneuron communication into temporally-distributed spikes, which reduce memory access frequency and data communication, and therefore, the computational power requirements of deep learning workloads. This becomes exceedingly important for continual data analysis in remote geographic regions which often lack reliable cloud access and power supply, where many wind power farms are stationed. This paper proposes an SNN architecture that achieves comparable performance with its ANN counterpart on a complex regression task, i.e., wind power interval prediction. The resulting multi-objective SNN demonstrates superior performance as compared with those from state-of-art ANNs in wind power interval prediction.