Short-term Quantitative Precipitation Forecasting (SQPF) using weather radar is an important but challenging problem as one must cope with inherent nonlinearity and spatiotemporal correlation in the data. In this paper, we propose a novel deep learning model, named Inductive spatiotemporal Graph Convolutional Networks (InstGCN), to overcome these issues in SQPF. The proposed InstGCN can learn a nonlinear mapping from historical radar reflectivity to future rainfall amounts, and extract informative spatiotemporal representations simultaneously. Specifically, we first provide a formal definition for formulating the SQPF problem from a graph perspective. Then, based on radar reflectivity and rain gauge observation, we propose a novel graph construction approach which utilizes a special elliptic structure to model the spatial dependency of precipitation area. Additionally, a new Node level Differential Block (Node-DB) is introduced to tackle the non-stationary temporal dependency. To execute inductive graph learning for unseen nodes, we design to decompose a whole graph into sub-graphs. We conduct extensive experiments on three real-world datasets in East China and a public weather radar dataset in the south-eastern parts of France. The experimental results confirm the advantages of InstGCN compared with several state-of-the-arts.