Multiferroics with ferroelectric and altermagnetic orders have been a focus in spintronics, for exploring emergent magnetoelectric coupling and electric-field controlled low-power spintronics. Using symmetry analysis and first-principles calculations, we verify that quasi-one-dimensional MoOBr3 is a ferroelectric altermagnet. Reversal of the electronic polarization preserves the band spin splitting in reciprocal space but flips the sign of the linear magnetoelectric coupling coefficient in real space, realizing a nonvolatile magnetoelectric response. Upon applying hydrostatic pressure, a structural phase transition occurs from the ferroelectric phase with parallel interchain polarizations to the antiferroelectric phase with antiparallel interchain polarizations. Taking into account spin-orbit coupling, in-plane rotation of the N´eel vector drives the system into a polar magnetic point group and induces ferroelectric polarization along the chain, enabling the strong magnetically driven magnetoelectric coupling characteristic. Our findings demonstrate that the synergy between ferroelectricity and altermagnetism enables nonvolatile and switchable magnetoelectric coupling in MoOBr3, providing a potential platform for energy-efficient magnetoelectric memory and electrically controlled magnetoelectric functionalities.