Colliding-wind binaries have long been thought to accelerate particles in the shocked wind. Recent detection of a hard $\gamma$-ray spectrum by the Fermi-LAT from $\eta$ Carinae hints to the presence of a hadronic component dominating in the $\approx 10$-300 GeV range, presumably from protons accelerated in the shocks and interacting ($pp$) with particles in the wind. Neutrinos are naturally produced in $pp$ interactions and emitted together with $\gamma$ rays. Detection of this multi-messenger signal can be very powerful to probe characteristics of the hadronic $\gamma$-ray component as well as particle accleration. We show that detection of high-energy neutrinos from $\eta$ Carinae by neutrino telescopes can probe the maximum shock-accelerated proton energy in the $\gtrsim 0.1$ PeV range.