The Purcell factor and the spontaneous emission spectrum of a quantum emitter (QE) placed close to the edge of a graphene half-space nanostructure is investigated, using semianalytical methods at the electrostatic regime. The half-space geometry supports an edge and a bulk surface plasmon (SP) mode. The Purcell factor of the QE is enhanced over eight orders of magnitude when its emission energy matches the resonance energy modes, for a specific value of the in-plane wave vector, at a separation distance of 10 nm. The different transition dipole moment orientations influence differently the enhancement factor of a QE, leading to large anisotropic behavior when positioned at different places above the half-space geometry. The field distribution is presented, showing clearly the excitation of the SP modes at the edge of the nanostructures. Also, we present the spontaneous emission spectrum of the QE near the half-space graphene nanostructure, and we show that strong light-matter coupling may emerge. When a QE with a free-space lifetime of 10 ns is placed at a distance of 10 nm from the edge of the graphene half-space, a Rabi splitting of 109 meV is found for emission energy of 0.15 eV. Our contribution can be used for designing future quantum applications using a combination of QEs and graphene nanostructures.