  In this paper, we consider backward stochastic differential equations driven
by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We
prove the existence and uniqueness of solution for such equations. On the one
hand, a priori estimates are obtained by applying the Girsanov type theorem in
the $G$-framework, from which we deduce the uniqueness. On the other hand, to
prove the existence of solutions, we first construct solutions for discrete
GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate
solutions for general quadratic GBSDEs in Banach spaces.
