The application of a Point-Collocation Non-Intrusive Polynomial Chaos method to the uncertainty quantification of a stochastic transonic aeroelastic wing problem has been demonstrated. The variation in the transient response of the first aeroelastic mode of a three-dimensional wing in transonic flow due to the uncertainty in free-stream Mach number and angle of attack was studied. A curve-fitting procedure was used to obtain time-independent parameterization of the transient aeroelastic responses. Among the uncertain parameters that characterize the time-dependent transients, the damping factor was chosen for uncertainty quantification, since this parameter can be thought as an indicator for flutter. Along with the mean and the standard deviation of the damping factor, the probability of having flutter for the given uncertainty in the Mach number and the angle of attack has been also calculated. Besides the Point-Collocation Non-Intrusive Polynomial Chaos method, 1000 Latin Hypercube Monte Carlo simulations were also performed to quantify the uncertainty in the damping factor. The results obtained for various statistics of the damping factor including the flutter probability showed that an 8th degree Point-Collocation Non-Intrusive Polynomial Chaos expansion is capable of estimating the statistics at an accuracy level of 1000 Latin Hypercube Monte Carlo simulation with a significantly lower computational cost. In addition to the uncertainty quantification, the response surface approximation, sensitivity analysis, and reconstruction of the transient response via Non-Intrusive Polynomial Chaos were also demonstrated.