Silicon-vacancy (SiV) color centers in diamond offer novel possibilities to probe light-matter interaction in nano-quantum optics and provide a scalable quantum system suitable for single-photon emission. However, their effective count-rate is still limited by non-radiative decay channels, radiation at wide angles and total internal reflection at the diamond interface. Recently optical antennas have been utilized in order to modify light-matter interaction at the nanoscale. Acting as resonators, they are able to increase the spontaneous emission rate of quantum emitters by several orders of magnitude in a broad spectral range. Because coupling a quantum emitter to a nano-antenna requires close proximity between the two systems, we implant Si ions on very thin diamond membranes that provide the required dimension for near-field interaction in a controlled manner and investigate the optical properties of SiV color centers in such diamond membranes. We consider gold nano-cones as nano-antennas, fabricated by focused electron beam induced deposition (EBID), followed by sputtering and focused ion beam (FIB) milling. The finite-difference time-domain (FDTD) calculations show that gold nano-cones can provide more than four orders of magnitude enhancement in the Purcell factor with an antenna efficiency (AE) of 80%.