In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasilinear parabolic partial differential equations (PDEs). The algorithms relies on a learning process by minimizing the path-wise difference of two discrete stochastic processes, which are defined by the time discretization of the FBSDEs and the DNN representation of the PDE solutions, respectively. The proposed algorithms demonstrate a convergence for a 100-dimensional Black--Scholes--Barenblatt equation at a rate similar to that of the Euler--Maruyama discretization of the FBSDEs.