  In this study, we extend the standard cosmological model within the quadratic
energy-momentum squared gravity (qEMSG) framework, introducing a nonminimal
interaction between the usual material field ($T_{\mu\nu}$) and its
accompanying quadratic energy-momentum squared field (qEMSF, $T_{\mu\nu}^{\rm
qEMSF}$), defined by $f(\mathbf{T}^2) = \alpha \mathbf{T}^2$ with
$\mathbf{T^2}=T_{\mu\nu}T^{\mu\nu}$. Focusing on high energy scales relevant to
big bang nucleosynthesis (BBN), we employ $^4$He abundance to constrain the
parameter $\alpha$. Our analysis selects the radiation-dominated universe
solution compatible with the standard cosmological model limit as $\alpha
\rightarrow 0$ and reveals that qEMSF interaction model can modify the
radiation energy density's evolution, potentially altering neutron-proton
interconversion rates and consequently affecting $^4$He abundance in various
ways. We establish the most stringent cosmological bounds on $\alpha$: $(-8.81
\leq \alpha \leq 8.14) \times 10^{-27} \, \mathrm{eV}^{-4}$ (68\% CL) from Aver
\textit{et al.}'s primordial $^4$He abundance measurements, aligning with
$\alpha=0$. Additionally, $(3.48 \leq\alpha \leq 4.43)\,\times 10^{-27}
\rm{eV}^{-4}$ (68\% CL) from Fields \textit{et al.}'s estimates, utilizing the
Planck-CMB estimated baryon density within the standard cosmological model
framework, diverges from $\alpha=0$, thereby lending support to the qEMSF
interaction model. The study also highlights the bidirectional nature of
energy-momentum/entropy transfer in qEMSF interaction model, depending on the
sign of $\alpha$. The implications of qEMSF in the presence of additional
relativistic relics are also explored, showcasing the model's potential to
accommodate deviations from standard cosmology and the Standard Model of
particle physics.
