Yes, the minimal transitive model of ZFC is pointwise definable.
The minimal transitive model of ZFC, known as the Shephardson-Cohen model, is the model $\langle L_\alpha,\in\rangle$, where the ordinal $\alpha$ is smallest such that this is a model of ZFC.
If $M$ is any transitive model of ZFC, then the constructible universe $L^M$ of that model will also be a model of ZFC and have the form $L_\gamma$, and so the minimal model $L_\alpha$ will be contained in it.
In this sense, the minimal model is not merely minimal, but also least---it is a subset of all other transitive models.
It is consistent with ZFC that there is no transitive model of ZFC. For example, in the minimal model $L_\alpha$ itself there is no element that is a transitive model of ZFC. So it is consistent with ZFC that the minimal model does not exist.
Theorem. The minimal transitive model of ZFC is pointwise definable.
Proof. Suppose that $L_\alpha$ is the minimal transitive model of ZFC. This model satisfies $V=L$ and consequently has a definable global well ordering of the universe. Using this, we see that there are definable Skolem functions. The set of definable elements of $L_\alpha$ is therefore closed under Skolem witnesses and is therefore an elementary substructure of $L_\alpha$. By condensation, this collection is isomorphic by the Mostowski collapse to a model $L_\beta$, which must be a model of ZFC, and which furthermore is pointwise definable, since we included only definable elements. Since $\beta\leq\alpha$, it follows by minimality that $\beta=\alpha$. And so $L_\alpha$ is pointwise definable. $\Box$