In this paper, a direct arbitrary Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) scheme is proposed for simulating compressible multi-material flows on the adaptive quadrilateral meshes. Our scheme couples a conservative equation related to the volume-fraction model with the Euler equations for describing the dynamics of the fluid mixture. The coupled system is discretized in the reference element and we use a kind of Taylor expansion basis functions to construct the interpolation polynomials of the variables. We show the property that the material derivatives of the basis functions in the DG discretization are equal to zero, with which the scheme is simplified. In addition, the mesh velocity in the ALE framework is obtained by using the adaptive mesh method from [H.Z. Tang and T. Tang, Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM J. NUMER. ANAL]. This adaptive mesh method can automatically concentrate the mesh nodes near the regions with large gradient values and greatly reduces the numerical dissipation near the material interfaces in the simulations. With the help of this adaptive mesh method, the resolution of the solution near the target regions can be greatly improved and the computational efficiency of the simulation is increased. Our scheme can be applied in the simulations for the gas and water media efficiently, and it is more concise compared to some other methods such as the indirect ALE methods. Several examples including the gas-water flow problem are presented to demonstrate the efficiency of our scheme, and the results show that our scheme can capture the wave structures sharply with high robustness.