  We prove a uniqueness result of the unbounded solution for a quadratic
backward stochastic differential equation whose terminal condition is unbounded
and whose generator $g$ may be non-Lipschitz continuous in the state variable
$y$, non-convex (non-concave) in the state variable $z$, and instead satisfies
a strictly quadratic condition and an additional assumption. The key
observation is that if the generator is strictly quadratic, then the quadratic
variation of the first component of the solution admits an exponential moment.
Typically, a Lipschitz perturbation of some convex (concave) function satisfies
the additional assumption mentioned above. This generalizes some results
obtained in \cite{BriandHu2006PTRF} and \cite{BriandHu2008PTRF}.
