We report inelastic neutron scattering results that reveal an hour-glass magnetic excitation spectrum in La_1.75Sr_0.25CoO ₄. The magnetic spectrum is similar to that observed previously in La_1.67Sr_0.33CoO₄, but the spectral features are broader. We show that the spectrum of La_1.75Sr _0.25CoO₄ can be modelled by the spin dynamics of a system with a disordered cluster spin glass ground state. Bulk magnetization measurements are presented that support the proposed glassy ground state. The observations reiterate the importance of quasi-one-dimensional magnetic correlations and disorder for the hour-glass spectrum, and suggest that disordered spin and charge stripes exist at lower doping in La _2-xSr_xCoO₄ than previously thought.