  In this paper, we first study one-dimensional quadratic backward stochastic
differential equations driven by $G$-Brownian motions ($G$-BSDEs) with
unbounded terminal values. With the help of a $\theta$-method of Briand and Hu
[4] and nonlinear stochastic analysis techniques, we propose an approximation
procedure to prove existence and uniqueness result when the generator is convex
(or concave) and terminal value is of exponential moments of arbitrary order.
Finally, we also establish the well-posedness of multi-dimensional G-BSDEs with
diagonally quadratic generators.
