In the present article, we introduce $k -$type slant helix and generalized Bertrand curve on $3 -$dimensional quadratic Lie group $G^3$. Specifically, we find necessary and sufficient conditions for any curve to become a $k -$type slant helix on $G^3.$ In addition, the relation between the Frenet frame of generalized Bertrand curve and the Frenet frame of generalized Bertrand mate, is established. Next, we find a condition for generalized Bertrand mate when the generalized Bertrand curve is $1 -$type slant helix in $G^3$. Further, we see the energy of the unit vector field generated by the axis of $k -$type slant helix.