We present a simple quark model for nucleus-nucleus collisions which employs the Glauber multiple scattering theory. The model is first used to fit a quark-quark scattering amplitude to high energy nucleon-nucleon elastic scattering data. This amplitude is then used to calculate high energy \ensuremath{\alpha}p and \ensuremath{\alpha}\ensuremath{\alpha} elastic scattering cross sections. The nucleus-nucleus calculations are performed using both a simple first-order optical phase shift function (essentially a t\ensuremath{\rho} approximation) and an optical phase shift function calculated through fifth order. We also calculate the same cross sections using a conventional nuclear model together with Glauber theory. Comparisons of the results of the different theoretical approaches and the data are presented.