We study intrinsic noise of current in a superconducting single-electron transistor, taking into account the combined effects of coherence and Coulomb interaction near a Cooper-pair resonance. The statistics of tunneling events is shown to be non-Poissonian. The zero-frequency noise is suppressed significantly when the quasiparticle tunneling rates are comparable to the coherent oscillation frequency of Cooper pairs. At frequencies close to the coherent oscillation frequency, the noise power exhibits strong quantum effects.