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Based on a couple of references, it seems that the answer is yes, see for example Boneta-Dierolf, 1992 and Bierstedt-Bonet, 1989.

However, from a comment to the answer of this MO question, I infer that there is a counterexample.

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Yes, they are. This should be in Grothendieck's Sur les espaces (F) et (DF). A good reference is the book Introduction to Functional Analysis of Meise and Vogt. Corollary 25.14 there says that reflexive Frechet spaces are distinguished (and, of course, Frechet Montel spaces are reflexive).

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