Low-energy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi/> <mml:mn>3</mml:mn> </mml:msup> </mml:math> He( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>α</mml:mi> </

\textbf{Background:} The $^{3}$He($α,γ$)$^{7}$Be reaction plays a crucial role in the proton-proton chain and Big Bang nucleosynthesis, affecting solar neutrino fluxes and primordial element abundances. Experimental data at astrophysical energies remain uncertain due to the extremely low cross sections. \\ \textbf{Purpose:} This work uses a microscopic potential-model approach to construct the $^{3}$He+$α$ potential from the nucleon+$α$ interaction, aiming to describe low-energy elastic scattering and to calculate the astrophysical $S$ factor of the $^{3}$He($α,γ$)$^{7}$Be reaction. \\ \textbf{Method:} The nucleon-nucleus potential is derived from self-consistent Skyrme Hartree-Fock (HF) calculations extended to the continuum. The $^{3}$He+$α$ potential is then obtained by folding the HF potential with the $^{3}$He density. A small number of scaling parameters is constrained by elastic-scattering data.\\ \textbf{Result:} The scaled Skyrme HF potential and folded potential simultaneously reproduce the low-energy $p$+$α$ and $^{3}$He+$α$ $s$-wave phase shifts, respectively. The calculated astrophysical $S$ factor of $^{3}$He($α,γ$)$^{7}$Be shows good agreement with experimental data, yielding the recommended value $S_{34}(0) = 0.610 \pm 0.024$~keV~b. A moderate sensitivity of $S_{34}(0)$ to the choice of projectile density is also observed in the folding procedure. \\ \textbf{Conclusion:} The Skyrme HF-based potential provides a unified and predictive microscopic framework for describing both elastic scattering and radiative capture in light nuclei.