We report the first measurement of the fraction of J/psi mesons coming from B-meson decay (F (B -> J/psi)) in p + p collisions at root s = 510 GeV. The measurement is performed using the forward silicon vertex detector and central vertex detector at PHENIX, which provide precise tracking and distance-of-closest-approach determinations, enabling the statistical separation of J=. due to B-meson decays from prompt J/psi. The measured value of F (B -> J/psi) is 8.1% +/- 2.3% (stat) +/- 1.9% (syst) for J/psi with transverse momenta 0 < p(T) < 5 GeV/c and rapidity 1.2 < vertical bar y vertical bar < 2.2. The measured fraction F (B -> J/psi) at PHENIX is compared to values measured by other experiments at higher center of mass energies and to fixed-order-next-toleading- logarithm and color-evaporation-model predictions. The b (b) over bar cross section per unit rapidity [d sigma/dy(pp -> b (b) over bar)] extracted from the obtained F (B -> J/psi) and the PHENIX inclusive J/psi cross section measured at 200 GeV scaled with color-evaporation-model calculations, at the mean B hadron rapidity y = +/- 1.7 in 510 GeV p + p collisions, is 3.63(-1.70)(+1.92) mu b. It is consistent with the fixed-order-next-to leading- logarithm calculations.