The computational prediction algorithm of neural network, or deep learning, has drawn much attention recently in statistics as well as in image recognition and natural language processing. Particularly in statistical application for censored survival data, the loss function used for optimization has been mainly based on the partial likelihood from Cox's model and its variations to utilize existing neural network library such as Keras, which was built upon the open source library of TensorFlow. This paper presents a novel application of the neural network to the quantile regression for survival data with right censoring, which is adjusted by the inverse of the estimated censoring distribution in the check function. The main purpose of this work is to show that the deep learning method could be flexible enough to predict nonlinear patterns more accurately compared to existing quantile regression methods such as traditional linear quantile regression and nonparametric quantile regression with total variation regularization, emphasizing practicality of the method for censored survival data. Simulation studies were performed to generate nonlinear censored survival data and compare the deep learning method with existing quantile regression methods in terms of prediction accuracy. The proposed method is illustrated with two publicly available breast cancer data sets with gene signatures. The method has been built into a package and is freely available at \url{https://github.com/yicjia/DeepQuantreg}.