We present a generalization of g-frames related to an adjointable operator $K$ on a Hilbert $C^{\ast}$-module, which we call $K$-g-frames. We obtain several characterizations of $K$-g-frames and we also give conditions under which the removal of an element from a $K$-g-frame leaves again a $K$-g-frame. In addition, we define a concept of dual, and using it we study the relation between a $K$-g-frame and a g-Bessel sequence with respect to different sequences of Hilbert $C^{\ast}$-modules.