This work demonstrates the efficient tuning of incoherent and coherent coupling between emitters embedded in an epsilon-near-zero (ENZ) waveguide coated with a multilayer graphene. As a result, a tunable two-qubit quantum phase gate based on the ENZ waveguide is realized at the cutoff frequency. Furthermore, due to the vanishingly small permittivity of the ENZ waveguide, all incoherent coupling between any two identical emitters located in the central area of the slit approaches a maximum, enabling near-ideal bipartite and multipartite entanglement. The coherent coupling between emitters is much larger at an operating frequency far from the ENZ resonance frequency than at the cutoff frequency, and the coherent coupling and resulting energy transfer efficiency can also be effectively tuned by the Fermi level of graphene. These results demonstrate an efficiently tunable electro-optical platform for quantum devices.