We report the first measurement of the fraction of $J/psi$ mesons coming from $B$-meson decay ($F_{B{rightarrow}J/psi}$) in $p$+$p$ collisions at $sqrt{s}=$ 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 $J/psi$ due to $B$-meson decays from prompt $J/psi$. The measured value of $F_{B{rightarrow}J/psi}$ is 8.1%$pm$2.3% (stat)$pm$1.9% (syst) for $J/psi$ with transverse momenta $0<p_T<5$ GeV/$c$ and rapidity $1.2<|y|<2.2$. The measured fraction $F_{B{rightarrow}J/psi}$ at PHENIX is compared to values measured by other experiments at higher center of mass energies and to fixed-order-next-to-leading-logarithm and color-evaporation-model predictions. The $bbar{b}$ cross section per unit rapidity ($dsigma/dy(pp{rightarrow}bbar{b})$) extracted from the obtained $F_{B{rightarrow}J/psi}$ and the PHENIX inclusive $J/psi$ cross section measured at 200 GeV scaled with color-evaporation-model calculations, at the mean $B$ hadron rapidity $y={pm}1.7$ in 510 GeV $p$$+$$p$ collisions, is $3.63^{+1.92}_{-1.70}mu$b, and it is consistent with the fixed-order-next-to-leading-logarithm calculations.
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