We perform magnetic susceptibility and magnetic torque measurements on the organic $kappa$-(BEDT-TTF)$_2$Hg(SCN)$_2$Br, which is recently suggested to host an exotic quantum dipole-liquid in its low-temperature insulating phase. Below the metal-insulator transition temperature, the magnetic susceptibility follows a Curie-Weiss law with a positive Curie-Weiss temperature, and a particular $Mpropto sqrt{H}$ curve is observed. The emergent ferromagnetically interacting spins amount to about 1/6 of the full spin moment of localized charges. Taking account of the possible inhomogeneous quasi-charge-order that forms a dipole-liquid, we construct a model of antiferromagnetically interacting spin chains in two adjacent charge-ordered domains, which are coupled via fluctuating charges on a Mott-dimer at the boundary. We find that the charge fluctuations can draw a weak ferromagnetic moment out of the spin singlet domains.
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