We develop a novel deep learning method for uncertainty quantification in stochastic partial differential equations based on Bayesian neural network (BNN) and Hamiltonian Monte Carlo (HMC). A BNN efficiently learns the posterior distribution of the parameters in deep neural networks by performing Bayesian inference on the network parameters. The posterior distribution is efficiently sampled using HMC to quantify uncertainties in the system. Several numerical examples are shown for both forward and inverse problems in high dimension to demonstrate the effectiveness of the proposed method for uncertainty quantification. These also show promising results that the computational cost is almost independent of the dimension of the problem demonstrating the potential of the method for tackling the so-called curse of dimensionality.