We report the detection of outflowing molecular gas in the center of the nearby (z = 0.014) massive radio galaxy NGC 6328. The radio core of the galaxy, PKS B1718-649, is identified as a gigahertz peaked spectrum source with a compact (2 pc) double radio lobe morphology. We used ALMA CO(2-1) and CO(3-2) observations at 100 pc resolution to study the gas kinematics up to ∼5 kpc from the galaxy center. While the bulk of the molecular gas is settled in a highly warped disk, in the inner 300 pc of the disk and along with the orientation of the radio jet, we identified high-excitation and high-velocity gas that cannot be attributed to any regular kinematic component based on our detailed 3D modeling of the ALMA data. The high-velocity dispersion in the gas also suggests that it is not part of an inflowing, shredding structure. These results suggest the presence of a molecular outflow of 3-8 solar masses per year. The outflow possibly originated from the interaction of the jet with the dense interstellar medium, even though the radio emission is detected closer to the center than the outflow. In this sense, this source resembles NGC 1377, 4C31.04, and ESO 420-G13, in which the outflows are linked to faint or past jet activity.