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The recent rapid growth in mobile data traffic entails a pressing demand for improving the throughput of the underlying wireless communication networks. Network node deployment has been considered as an effective approach for throughput enhancement which, however, often leads to highly non-trivial non-convex optimizations. Although convex approximation based solutions are considered in the literature, their approximation to the actual throughput may be loose and sometimes lead to unsatisfactory performance. With this consideration, in this paper, we propose a novel graph neural network (GNN) method for the network node deployment problem. Specifically, we fit a GNN to the network throughput and use the gradients of this GNN to iteratively update the locations of the network nodes. Besides, we show that an expressive GNN has the capacity to approximate both the function value and the gradients of a multivariate permutation-invariant function, as a theoretic support to the proposed method. To further improve the throughput, we also study a hybrid node deployment method based on this approach. To train the desired GNN, we adopt a policy gradient algorithm to create datasets containing good training samples. Numerical experiments show that the proposed methods produce competitive results compared to the baselines.