A magnetic order can be completely suppressed at zero temperature (T), by doping carriers or applying pressure, at a quantum critical point, around which physical properties change drastically. However, the situation is unclear for an electronic nematic order that breaks rotation symmetry. Here, we report nuclear magnetic resonance studies on NaFe1-xCoxAs where magnetic and nematic transitions are well separated. The nuclear magnetic resonance spectrum is sensitive to inhomogeneous magnetic fields in the vortex state, which is related to London penetration depth λL that measures the electron mass m*. We discovered two peaks in the doping dependence of λ^2L (T~0T), one at Xm=0.027 where the spin-lattice relaxation rate shows quantum critical behavior, and another at Xc=0.032 around which the nematic transition temperature extrapolates to zero and the electrical resistivity shows a T-linear variation. Our results indicate that a nematic quantum critical point lies beneath the superconducting dome at xc where m* is enhanced. The impact of the nematic fluctuations on superconductivity is discussed.