  In this work, we consider the incompressible generalized Navier-Stokes-Voigt
equations in a bounded domain $\mathcal{O}\subset\mathbb{R}^d$, $d\geq 2$,
driven by a multiplicative Gaussian noise. The considered momentum equation is
given by:
  \begin{align*}
  \mathrm{d}\left(\boldsymbol{u} - \kappa \Delta \boldsymbol{u}\right) =
\left[\boldsymbol{f} +\operatorname{div}
\left(-\pi\mathbf{I}+\nu|\mathbf{D}(\boldsymbol{u})|^{p-2}\mathbf{D}(\boldsymbol{u})-\boldsymbol{u}\otimes
\boldsymbol{u}\right)\right]\mathrm{d} t + \Phi(\boldsymbol{u})\mathrm{d}
\mathrm{W}(t).
  \end{align*} In the case of $d=2,3$, $\boldsymbol{u}$ accounts for the
velocity field, $\pi$ is the pressure, $\boldsymbol{f}$ is a body force and the
final term stay for the stochastic forces. Here, $\kappa$ and $\nu$ are given
positive constants that account for the kinematic viscosity and relaxation
time, and the power-law index $p$ is another constant (assumed $p>1$) that
characterizes the flow. We use the usual notation $\mathbf{I}$ for the unit
tensor and $\mathbf{D}(\boldsymbol{u}):=\frac{1}{2}\left(\nabla \boldsymbol{u}
+ (\nabla \boldsymbol{u})^{\top}\right)$ for the symmetric part of velocity
gradient. For $p\in\big(\frac{2d}{d+2},\infty\big)$, we first prove the
existence of a martingale solution. Then we show the pathwise uniqueness of
solutions. We employ the classical Yamada-Watanabe theorem to ensure the
existence of a unique probabilistic strong solution.Then we show the pathwise
uniqueness of solutions. We employ the classical Yamada-Watanabe theorem to
ensure the existence of a unique probabilistic strong solution.
