Boron-nitride (BN) domains in graphene or graphene domains in BN monolayer offer additional freedoms for tuning the electronic properties of these BN/C nanostructures, which is quite crucial for the applications in nanoscale devices. Based on first-principles calculations combined with a simple Hubbard model, we show that the electron zero-energy states (ZESs) of BN/graphene core-shell quantum dots (QDs) in triangular shapes can be well tuned by varying the size and topology of each domain. The net spin of the systems is dominated by the graphene segment which can be described by a Lieb's theorem. We also demonstrated that a π-electron Hubbard model within a mean-field approximation is implementable in dealing with the electron spin-polarization of BN/C hetero-structured graphene-like materials. This provides an efficient theoretical approach for the BN/C systems where electron spin-polarization is involved.