  Binary neutron-star (BNS) mergers are accompanied by multi-messenger
emissions, including gravitational wave (GW), neutrino, and electromagnetic
signals. Some fraction of BNS mergers may result in a rapidly spinning magnetar
as a remnant, which can enhance both the EM and neutrino emissions. In this
study, we model the neutrino emissions from such systems and discuss the
prospects for detecting the neutrinos coincident with GW signatures. We
consider a scenario where a magnetar remnant drives a pulsar wind using its
spin energy. The wind interacts with the surrounding kilonova ejecta, forming a
nebula filled with non-thermal photons. Ions and nuclei extracted from the
magnetar's surface can be accelerated in the polar-cap and the
termination-shock regions. We investigate the neutrino fluences resulting from
photomeson interactions, where accelerated CR protons interact with the photons
in the nebula. Our findings indicate that the peak neutrino fluence is $\sim
10^{-2}\rm GeV~cm^{-2}$ for a source at $40$ Mpc, which is reached
approximately $\mathcal{O}( 1-10\ {\rm days})$ post merger. Finally, we examine
the potential for GW-triggered stacking searches with IceCube-Gen2 using
next-generation GW detectors such as the Cosmic Explorer (CE) and the Einstein
Telescope (ET). We conclude that, assuming an optimistic neutrino emission
model, a combination of CE+ET would offer a high probability of neutrino
detection from these sources within an operational timescale of $\sim 20$
years. In case of non-detection, $2 \sigma$ level constraints on model
parameters can be established within similar joint operation timescales.
