Recently, the cubic Galileon gravity (CGG) model has been suggested as an alternative gravity theory to general relativity. The model consists of an extra field potential term that can serve as the "fifth force." In this article, we examine the possibility of whether or not this extra force term can explain the missing mass problem in galaxies without the help of dark matter. By using the Milky Way rotation curve and the Spitzer Photomery and Accurate Rotation Curves data, we show that this CGG model can satisfactorily explain the shapes of these rotation curves without dark matter. The CGG model can be regarded as a new alternative theory to challenge the existing dark matter paradigm. Copyright © 2018 The American Astronomical Society. All rights reserved.