We explore the photoluminescence (PL) properties of hexagonal boron nitride (h-BN) quantum emitters embedded within atomically thin graphene/h-BN heterostructures fabricated by mechanical transfer. Stable light emission could be observed from h-BN emitters which due to the local presence of multilayer h-BN are not subject to fluorescence quenching by graphene. By using graphene as a top gate contact, the PL emission can be tuned by up to 24 meV per V/nm, with a high robustness of the emitters over several voltage sweep cycles. Two different types of h-BN emitters were observed, one with a quadratic and the other one with a linear Stark shift. Moreover, the vertical electric field leads to an asymmetric modulation of both the fluorescence intensity and lifetime between the negative and positive gate voltage regimes. The overall behavior can be well explained by a model involving different rates for electron and hole tunneling between the h-BN and graphene layers. Our findings suggest ultrathin van der Waals heterostructures as valuable platforms for fine tuning the optoelectronic properties of atomic defect-based quantum emitters. Published under license by AIP Publishing.