Tracing signatures of modified gravity in redshift-space galaxy bispectrum multipoles: Prospects for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>E</mml:mi> <mml:mi>u</mml:mi> <mml:mi>c</mml:mi> <mml:mi>l</mml:mi

We study the galaxy bispectrum multipoles in the Hu-Sawicki $f(R)$ gravity model, where a scalar degree of freedom mediates a fifth force that is screened in high-density environments. The model is specified by $f_{R0}$, the present-day background value of the scalar field, which controls the strength of deviations from General Relativity (GR). Using perturbation theory, we compute the redshift-space galaxy bispectrum with the full scale- and time-dependent second-order kernels, incorporating corrections from the scale-dependent growth rate and nonlinear screening. Expanding the bispectrum in spherical harmonics, we analyze the sensitivity of the multipoles to modified gravity and forecast their detectability in a \textit{Euclid}-like survey. The monopole ($B_0^0$) and quadrupole ($B_2^0$) show the strongest signatures, with relative deviations of $2\%$--$8\%$ at $z=0.7$ and $k_1\simeq0.3\,h\,{\rm Mpc}^{-1}$ (largest side of the triangle) for $f_{R0}=10^{-5}$. Higher multipoles provide weaker but complementary signals. For \textit{Euclid}, we forecast signal-to-noise ratios up to $\sim30$ for the monopole and $\sim15$ for the quadrupole including the Finger-of-God damping and shot noise effect. These results demonstrate that bispectrum multipoles are a powerful probe of gravity, capable of breaking degeneracies with bias and velocity effects and strengthening constraints on deviations from $Λ$CDM.