Suppose we are in ZFC, let $\mathcal U$ be an uncountable Grothendieck universe and consider the set of its parts $\mathcal{P(U)}$. (I will index axioms as $(\mathcal U.n)$)
Note that if $x \in \mathcal U$, then $x \in\mathcal{P(U)}$, directly from $(\mathcal U.1)$ (i.e., $\mathcal U$ is a transitive set).
Further, if $x \in \mathcal{P(U)}$ and $|x|<|\mathcal U|$, then $x \in \mathcal U$. (This is not so trivial and require the axiom of choice. Actually, in ZFC, Grothendieck universes are equivalent to the so called transitive Tarski class; a sketch of the proof, in italian, can be found here).
Thus, $\mathcal{P}_<(\mathcal U) := \{x \in \mathcal{P(U)}\,|\,|x|<|\mathcal U|\} = \mathcal U$.
Denoting $\mathcal P_=(\mathcal U) := \{C \in \mathcal{P(U)}\,|\,|C|=|\mathcal U|\}$, we have $\mathcal{P(U)} = \mathcal{P}_<(\mathcal U) \cup \mathcal P_=(\mathcal U) = \mathcal U \cup \mathcal P_=(\mathcal U)$, the union being disjoint.
Define the binary relation $\in_{\mathcal U} \subseteq \mathcal{P(U)}\times \mathcal{P(U)}$ such that ${\in_{\mathcal U}}_{|\mathcal U \times \mathcal{P(U)}} = {\in_{\text{ZFC}}}_{|\mathcal U \times \mathcal{P(U)}}$ and ${\in_{\mathcal U}}_{|\mathcal P_=(\mathcal U) \times \mathcal{P(U)}} = \emptyset$. That is, $\in_{\mathcal U}$ restricted to $\mathcal U \times \mathcal{P(U)}$ is just the usual set-relation $\in_{\text{ZFC}}$ of ZFC (i.e., just $\in$, if you prefer); and there is no other couple in $\in_{\mathcal U}$.
Does $(\mathcal{P(U)}, \in_{\mathcal U})$ serve as a model for NBG?
As two-sorted theory, this is equivalent to choose $\mathcal{P(U)}$ as classes, as sets the parts $x \in \mathcal{P(U)}$ such that $x \in \mathcal U$ (i.e., the parts $x \subset U$ such that $|x| < |\mathcal U|$) and as binary relation the modified membership $\in_{\mathcal U}$ defined above.
Yes, the power set of a Grothendieck universe is a model of NBG, and even of the stronger Morse-Kelley set theory.