In this work, we investigate pressure-driven water flows in graphene-coated copper nanochannels through molecular dynamics simulations. It is found that the flow rate in bare copper nanochannel can be significantly enhanced by a factor of 45 when the nanochannel is coated with monolayer graphene. The enhancement factor for the flow rate reaches about 90 when the nanochannel is modified with 3 or more graphene layers. The dipole relaxation time and the hydrogen bond lifetime of interfacial water molecules show that the graphene coating promotes the mobility of water molecules at the interface. The distribution of the potential of mean force and the free energy barriers also confirm that graphene coating reduces the flow resistance and 3 layers of graphene can fully screen the surface effects. The results in this work provide important information for the design of graphene-based nanofluidic systems for flow enhancement.