  Focusing on the local type primordial non-Gaussianities, we study the
bispectrum and trispectrum during a non-minimal slow-roll inflation. We use the
so-called $\delta N$ formalism to investigate the super-horizon evolution of
the primordial perturbations in this setup. Firstly we obtain the main
equations of the model and introduce the framework of the $\delta N$ formalism
for this case. Then we give analytical expressions for the nonlinear parameters
describing the non-Gaussianity in the slow-roll approximation. We analyze the
bispectrum by its non-linear parameter, $f_{NL}$. Furthermore, we calculate
$\tau_{NL}$ and $g_{NL}$ which are non-linear parameters characterizing the
amplitude of trispectrum. Finally, by adopting a quadratic form for both the
potential and non-minimal coupling (NMC) function, we test our setup in the
light of Planck2015 data and constrain the model parameters space. Although the
non-Gaussianity parameters are so small in this setup, this model is consistent
with recent observation. We extend our analysis to see the situation in the
Einstein frame and compare the results in these two frames.
