This paper is devoted to the introduction of a new class of consistent estimators of the fractal dimension of locally self-similar Gaussian processes. These estimators are based on linear combinations of empirical quantiles (L-statistics) of discrete variations of a sample path over a discrete grid of the interval [0,1]. We derive the almost sure convergence for these estimators and prove the asymptotic normality. The key-ingredient is a Bahadur representation for empirical quantiles of non-linear functions of Gaussians sequences with correlation function decreasing hyperbollically.