This paper extends a moving screw dislocation theory in a conventional crystal to a one-dimensional hexagonal quasicrystal in the framework of the Landau theory. By introducing two functions ϕ and ψ which satisfy wave equations, respectively, a general solution is suggested in terms of ϕ and ψ. The analytical expressions for displacement and stress fields induced by a moving screw dislocation as well as the energy are found. The results obtained here when imposing the condition that the phason fields is absent reduce to those for a moving screw dislocation in a crystal.