We conducted an investigation into the possible correlation between very-high-energy (VHE) gamma-ray sources, as identified by the Large High Altitude Air Shower Observatory (LHAASO), and high-energy neutrino events captured by the IceCube Neutrino Observatory. For this analysis, we utilized the First LHAASO catalog, which comprises 90 VHE gamma-ray sources, and the IceCube Event Catalog of Alert Tracks (IceCat-1), which includes 348 track-like neutrino events. Through Monte Carlo simulations, we assessed the correlations between the neutrino events and LHAASO sources, achieving the best p-values of 0.237, 0.312, and 0.115, respectively, for all LHAASO sources, KM2A sources, and WCDA sources. These findings imply that the spatial coincidence between LHAASO sources and neutrino alerts is relatively weak. Furthermore, under the assumption that gamma-rays are exclusively generated through hadronic interactions, we estimated the most optimistic theoretical neutrino fluxes from LHAASO sources and compared them with the IceCube observed fluxes. The results indicate that the majority of these sources are improbable counterparts to the detected neutrinos. In conclusion, LHAASO sources could be potential galactic neutrino emitters but currently do not exhibit compelling evidence, thereby underscoring the necessity for elevated sensitivity levels in forthcoming detection instruments like IceCube-Gen2.