Based on a couple of references, it seems that the answer is yes, see for example
[Boneta-Dierolf, 1992](http://www.sciencedirect.com/science/article/pii/S0304020808703207) and [Bierstedt-Bonet, 1989](http://dmle.cindoc.csic.es/revistas/detalle.php?numero=592).


However, from a comment to the answer of [this MO question](https://mathoverflow.net/questions/8443/barrelled-bornological-ultrabornological-semi-reflexive-how-are-these-us), I infer that there is a counterexample.



Is a Fréchet Montel space distinguished?
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).


