We present high-resolution images of millimeter-wave 115 GHz ^12^CO(1-0) and near-infrared 2.1213 micron H_2_ line emission and K-band continuum emission from the Wolf-Rayet nucleus of the low-ionization nuclear emission-line region (LINER) NGC 6764. We also present observations of 2.0581 micron He I and 2.1655 micron Brγ line emission, optical spectroscopic measurements of Wolf-Rayet features, and the first detection of NGC 6764 in X-rays. Our measurements reveal a dense concentration of molecular gas and a starburst at the nucleus of NGC 6764. A detailed starburst analysis indicates a recent intense starburst of a few times 10^7^ yr. This starburst is most likely responsible for the presence of a few thousand Wolf-Rayet stars in the nucleus of NGC 6764. An interferometric position-velocity map of the nucleus reveals the presence of distinct molecular cloud complexes with an apparently asymmetric velocity field shifted toward the blue with respect to the systemic velocity of 2420 km s^-1^. The distribution of the 2.12 micron H_2_ line flux exhibits extensions approximately perpendicular to the stellar bar that are in agreement with structural features in VLA radio maps and IRAM 30 m maps of the ^12^CO(2-1) line emission, possibly indicating a nuclear outflow. A detailed investigation of the Wolf-Rayet feature at 466 nm indicates that it is spatially extended on a scale of a few arcseconds.