We analyze strongly lensed images in 8 galaxy clusters to measure their dark matter density profiles in the radial region between 10 kpc and 100 kpc, and use this to constrain the self-interaction cross section of dark matter (DM) particles. We infer the mass profiles of the central DM halos, bright central galaxies, key member galaxies, and DM subhalos for the member galaxies for all 8 clusters using the QLens code. The inferred DM halo surface densities are fit to a self-interacting dark matter (SIDM) model, which allows us to constrain the self-interaction cross section. When our full method is applied to mock data generated from two clusters in the Illustris-TNG simulation, we find results consistent with no dark matter self-interactions as expected. For the eight observed clusters with maximum circular velocities in the range of 1400 to 2000 km/s, we find that the upper limit for the self-interaction cross section per unit mass of dark matter is 0.065 $cm^2/g$ at the 95% confidence level.