I tried to quickly Google and search for this here, but couldn't find anything relevant. Basically I want to know whether there is any common word for Japanese conversation; e.g. in the context of speaking about a Japanese conversation course, or otherwise.
I think 英会話 is special in many way. For example, 仏会話 or 独会話 aren't really used. For other language courses, you can use [language name] + 会話, e.g. フランス語会話 or ドイツ語会話 or 中国語会話, or even 外国語会話 in general. (Of course 英 is already special in that other language names are usually formed as [country name] + 語.)
英会話 probably meant "conversation in English" initially, but it has become almost synonymous with courses/classes teaching English conversational skills. Of course English is also special in that it is (at least statistically speaking) the only foreign language taught; and English is taught with an infamously poor focus on conversational skills, necessitating 英会話 classes.
I don't think the same is necessarily true for other languages, so to describe a French conversation class, you might want to explicitly say フランス語会話コース (or レッスン).
I guess that 日本語会話 is something not really taught to native speakers, so you might just use a more generic "Japanese language course" instead, for example 日本語講座, 日本語集中コース, 日本語勉強会, or whatever best describes your class. If it's a class really focussing on conversational skills, then maybe 日本語会話コース (or レッスン).