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In this paper, we first study one-dimensional quadratic backward stochasticdifferential equations driven by $G$-Brownian motions ($G$-BSDEs) withunbounded terminal values. With the help of a $\theta$-method of Briand and Hu[4] and nonlinear stochastic analysis techniques, we propose an approximationprocedure 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 withdiagonally quadratic generators.

Quadratic $G$-BSDEs with convex generators and unbounded terminal conditions