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Let $R$ be a regular, local $\mathbb{Q}$-algebra with a regular system of parameters $x_1, \dotsc, x_n$, and let $$f \colon \mathbb{Q}[X_1, \dotsc, X_n]_{(X_1, \dotsc, X_n)} \rightarrow R$$ be the map given by $X_i \mapsto x_i$. Then $f$ is flat (for instance, by Bourbaki, cf. EGA III, 0.10.2.2).

Is $f$ a regular morphism, that is, are the (geometric) fibers of $f$ regular? Certainly, the closed fiber is regular, but how about the others?

The answer should be yes, and I would appreciate an argument for this. In the case when $R$ is excellent, the positive answer seems to be a special case of EGA IV, 7.9.8, but a less contrived argument (and one that would not use an additional excellence assumption) would be greatly appreciated.

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You do not need $R$ excellent. Since $\mathbb{Q}[X_1, \dotsc, X_n]_{(X_1, \dotsc, X_n)}$ is excellent it follows from M. André, Localisation de la lissité formelle.

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