Recurrence plots of time series generated by discrete fractional Gaussian noise (fGn) processes are analyzed. We compute the probabilities of occurrence of consecutive recurrence points forming diagonals and verticals in the recurrence plot constructed without embedding. We focus on two recurrence quantification analysis measures related to these lines, respectively, the percent determinism and the laminarity (LAM). The behavior of these two measures as a function of the fGn’s Hurst exponent $H$ is investigated. We show that the dependence of the laminarity with respect to $H$ is monotonic in contrast to the percent determinism. We also show that the length of the diagonal and vertical lines involved in the computation of percent determinism and laminarity has an influence on their dependence on $H$. Statistical tests performed on the $LAM$ measure support its utility to discriminate fGn processes with respect to their $H$ values. These results demonstrate that recurrence plots are suitable for the extraction of quantitative information on the correlation structure of these widespread stochastic processes.