Multiple Fano resonances (FRs) can be produced by destroying the symmetry of structure or adding additional nanoparticles without changing the spatial symmetry, which has been proved in noble metal structures. However, due to the disadvantages of low modulation depth, large damping rate, and broadband spectral responses, many resonance applications are limited. In this research paper, we propose a graphene plasmonic metamolecule (PMM) by adding an additional 12 nanodiscs around a graphene heptamer, where two Fano resonance modes with different wavelengths are observed in the extinction spectrum. The competition between the two FRs as well as the modulation depth of each FR is investigated by varying the materials and the geometrical parameters of the nanostructure. A simple trimer model, which emulates the radical distribution of the PMM, is employed to understand the electromagnetic field behaviors during the variation of the parameters. Our proposed graphene nanostructures might find significant applications in the fields of single molecule detection, chemical or biochemical sensing, and nanoantenna.