For centrosymmetric materials such as monolayer graphene, no optical second-harmonic generation (SHG) is generally expected, because it is forbidden under the electric-dipole approximation. Yet we observe a strong, doping-induced SHG from graphene, with its highest strength comparable to the electric-dipole-allowed SHG in noncentrosymmetric 2D materials. This novel SHG has the nature of an electric-quadrupole response, arising from the effective breaking of inversion symmetry by optical dressing with an in-plane photon wave vector. More remarkably, the SHG is widely tuned by carrier doping or chemical potential, being sharply enhanced at Fermi-edge resonances but vanishing at the charge neutral point that manifests the electron-hole symmetry of massless Dirac fermions. This striking behavior in graphene, which should also arise in graphenelike Dirac materials, expands the scope of nonlinear optics and holds the promise of novel optoelectronic and photonic applications.