Laser driven positron source has the advantages of high yield, short pulse width and high energy. In this paper, particle-in-cell simulation and Monte-Carlo simulation are combined to simulate the process of positron production in the interaction of relativistic femtosecond laser with a micro-structured surface target (MST) with a micron-scale wire array on the surface. The results show that when the laser energy is about 6 J and the pulse width is about 40 fs, fast electrons with the yield of 10¹¹ orders of magnitude and the cut-off energy of about 120 MeV can be obtained. When the electrons bombard a high-Z conversion target, positrons with the yield of 10⁹ orders of magnitude, and cut-off energy about 50 MeV are obtained. The divergence angle of the positron beam is 4.92°. Compared with planar targets, the use of MSTs can benefit the yield, energy and directivity of positrons.