We use the statistics of caustic crossings induced by microlensing in the lens system Q 2237+0305 to study the lens galaxy peculiar velocity. We calculate the caustic crossing rates for a comprehensive family of stellar mass functions and find a dependence of the average number of caustic crossings with the effective transverse velocity and the average mass, < n > \propto {veff / \sqrt{< m > }}, equivalent to the theoretical prediction for the case of microlenses with identical masses. We explore the possibilities of the method to measure v eff using the ~12 yr of Optical Gravitational Lensing Experiment monitoring of the four images of Q 2237+0305. To determine a lower limit for v eff, we count, conservatively, a single caustic crossing for each one of the four high magnification events identified in the literature (plus one additional proposed by us) obtaining veff ≳ 240\sqrt{< m > /0.17 M_ȯ } km s-1 at 68% of confidence. From this value and the average FWHM of the four high magnification events, we obtain a lower limit of rs ≳ 1.4 \sqrt{< m > /0.17 M_ȯ } light-days for the radius of the source (rs = FWHM/2.35). Tentative identification of three additional caustic crossing events leads to estimates of veff≃ (493+/- 246)\sqrt{< m > /0.17 M_ȯ } km s-1 for the effective transverse velocity and of rs ≃ (2.7+/- 1.3)\sqrt{< m > /0.17 M_ȯ } light-days for the source size. The estimated transverse peculiar velocity of the galaxy is vt ≃ (429+/- 246)\sqrt{< m > /0.17 M_ȯ } km s-1.