It is shown that the redshifts z_o of γ-ray burst (GRB) sources, where they have been measured, together with the redshifts for seven quasars (QSOs) that lie very close to the positions of the unidentified sources GRB 990625, 000210, 001105 (two QSOs), 940720, 991217, and 990506, show a remarkable tendency to cluster about several of the periodic redshift peaks previously established for QSOs at z = 0.061, 0.30, 0.60, 0.96, 1.41, 1.96, 2.63, 3.44, and 4.45. In 1971, Karlsson showed that these peaks lie in a series with Δ log(1 + z) = 0.089. Out of a total of 32 currently known redshifts of GRBs, afterglows, or QSOs very close to burst positions, two are very close to 0.30, three are close to 0.60, nine are equal to or very close to 0.96, three are very close to 1.41, six are close to 1.96, two are close to 3.44, and one is very close to 4.45. Statistical tests by W. Napier show that the observed redshifts z_o showed periodicity at the 98% confidence level. In addition, very close to the positions of two bursts GRB 990625 and GRB 001105, many QSOs with redshifts close to the peak values have been found. Since z_o = [(1 + z_c)(1 + z_D)(1 + z_i) - 1], where z_c, z_D, and z_i are the cosmological, Doppler, and intrinsic components of the observed redshift z_o, the existence of these peaks suggests that z_o z_i, so that both z_c and z_D are very much less than z_o. However, while the observed values of z_o are very close to the corresponding values of z_i, in most cases z_o > z_i, suggesting that in most cases z_c is greater than it was found to be in earlier samples of X-ray QSOs that appear to be ejected from bright galaxies. It appears likely, therefore, that the GRB sources, like the QSOs, are ejected from active galaxies, most of which have comparatively small cosmological redshifts 0.02 ≤ z_c ≤ 0.1, thus suggesting that the distances of most of the GRB sources are ≤500 Mpc. A possible example of an active galaxy that has given rise to such phenomena is UGC 12348 (z_i = 0.03). This galaxy has two GRB sources and three QSOs with measured redshifts z_o that lie very close to intrinsic redshift peaks all lying within 1° of it. Among these five objects, the QSO at z = 3.7 lies inside the error box for the unidentified burst GRB 991217.