Armenian language help

The issue I have is that the only font that support that language in InDesign is Noto San Armenian but it does not support any characters like !:.?.

There is a way to apply your Noto font to only the Armenian stuff, leaving all of the shared punctuation in a different font. But you don't need to, because there are quite a few fonts that support Armenian! But Noto is the only one you have installed. I'm guessing you are on a Mac, right? Those are fonts installed in macOS to give you basic multilingual support. However, I have met exactly zero persons who use languages written in complex scripts who tell me that Noto fonts look good. I mean, doesn't it look like an off-brand Arial to you? 

So: you should go find some good Armenian fonts, then you won't have to figure out how to make a paragraph style that applies Noto Sans Armenian to some glyphs and some other catchall font to the question marks and suchlike. If you're interested, the secret is to make a base paragraph style in your base font, e.g. Myriad, and then make GREP style that applies a character style with Noto Sans Armenian only to fonts that have Unicode values in the Armenian range, like [\u{0530}-\u{058F}]+

BUT 

you probably aren't going to want to do that, because it'd be so much easier to use a good font, that had both Armenian glyphs as well as punctuation. You can head over to fonter.am and pick out a font or three. I have used "GHEA Mariam" in the past, which is one of the fonts that was developed for the Armenian government. There's a bunch of other fonts there as well, but most of them will have complete font complements and which are not, like Noto, designed to be the "font of last resort" and therefore missing glyphs likely found in other fonts.