Would concepts like grammar have even been understood/discussed until other languages with different grammars were encountered?
There are a couple of reasons to think that development of linguistics is not dependent on knowledge that languages differ. First, the first known tradition of grammatical description, developed in ancient India, came about in order to perfectly preserve the pronunciation of Vedic hymns, and was not in response to encountering other languages. Second, the underpinnings of western grammatican tradition are the writings of Aristotle (in Organon and Rhetoric), who developed a fairly sophisticated theory of language structure as a philosophical exercise. The originators of these traditions probably had some knowledge that there were languages other than Greek or Sanskrit, but there is no indication that the resulting theories had anything to do with the existence of such languages.
I would argue that, yes, grammar and linguistic studies would have been discussed even without contact with another language. Mainly because if the variation that one can find within a single language.
Today most of us are used to a more or less standardised language. It may be clear that language politics have made it so. Implicitly normative institutions such as dictionaries or explicitly normative ones such as the Dutch-Belgian Taalunie make sure that a language is documented and that some sort of standard is followed. Oftentimes this is an item of interest for government as well: if there is a lot of diversity within the same language in a country (regiolects, dialects), that could influence how some groups behave to each other and to the reigning powers. In other words, a very clear difference in dialect may cause (in extremis) political and social unrest. Culturally, variation is a treasure, but politically it makes things harder than they have to be.
To document and standardize a language, a grammar needs to be developed and some linguistic research is needed. Interestingly, this also leads you into the field of dialectology and not only to normative grammar studies.
In conclusion: yes, in a country where a written system was present but where spoken variation was common, we would still have developed grammar simply because of the ease of having a standard form of a language.