We propose a method to detect possible non-stationarities of gamma-ray burst jets. Assuming that the dominant source of variability in the prompt gamma light curve is the non-stationarity of the jet, we show that there should be a connection between the variability measure and the characteristic angle of the jet derived from the jet break time of the afterglow. We carried out Monte Carlo simulations of long gamma-ray burst observations assuming three radial luminosity density profiles for jets and randomizing all burst parameters, and created samples of gamma light curves by simulating jets undergoing Brownian motions with linear restoring forces. We were able to demonstrate that the connection between the variability and the characteristic angle is an anticorrelation in case of uniform and power-law jet profiles, and a correlation in case of a Gaussian profile. We have found that as low as 50 (144) gamma-ray burst observations with jet angle measurements can be sufficient for a 3σ (5σ) detection of the connection. The number of observations required for the detection depends on the underlying jet beam profile, ranging from 50 (144) to 237 (659) for the four specific profile models we tested.