In this paper we report the elliptic ($v_{2}$) and triangular ($v_{3}$) flow measurements of strange and multi-strange hadrons such as $K_{S}^{0}$, $\Lambda$($\bar{\Lambda}$), $\phi$, $\Xi^{-}$($\bar{\Xi}^{+}$), and $\Omega^{-}$($\bar{\Omega}^{+}$) at mid-rapidity ($|y| < 1.0$) in Au+Au collisions at $\sqrt{s_{NN}}$ = 19.6 GeV using high statistics Beam Energy Scan Phase-II (BES-II) data. The transverse momentum ($p_{T}$) dependence, centrality dependence, and the number of constituent quarks (NCQ) scaling of $v_{2}$ and $v_{3}$ are studied for all these particles. A better NCQ scaling is observed in the case of antiparticles compared to particles, which can be attributed to the contribution from the transported quarks in particles. The hydrodynamically motivated ratio ($v_{3}/v_{2}^{3/2}$) is presented as a function of $p_{T}$.