We report a measurement of the strange axial coupling constant $g_A^s$ using atmospheric neutrino data at KamLAND. This constant is a component of the axial form factor of the neutral-current quasielastic (NCQE) interaction. The value of $g_A^s$ significantly changes the ratio of proton and neutron NCQE cross sections. KamLAND is suitable for measuring NCQE interactions as it can detect nucleon recoils with low-energy thresholds and measure neutron multiplicity with high efficiency. KamLAND data, including the information on neutron multiplicity associated with the NCQE interactions, makes it possible to measure $g_A^s$ with a suppressed dependence on the axial mass MA, which has not yet been determined. For a comprehensive prediction of the neutron emission associated with neutrino interactions, we establish a simulation of particle emission via nuclear deexcitation of 12C, a process not considered in existing neutrino Monte Carlo event generators. Energy spectrum fitting for each neutron multiplicity gives $g_A^s$ =-0.14$_{-0.26}^{+0.25}$, which is the most stringent limit obtained using NCQE interactions without MA constraints. The two-body current contribution considered in this analysis relies on a theoretically effective model and electron scattering experiments and requires future verification by direct measurements and future model improvement.