I'm not aware of a way to do this. I think the rationale is like so: the document you are making has no dependency on RStudio itself, but only on the packages used in your code and the rmarkdown/knitr toolchain. I suspect that everything in this thread is still true:
https://support.rstudio.com/hc/en-us/community/posts/203910987-versionInfo-not-working
That being said, I can imagine that perhaps your document describes some sort of workflow that involves RStudio and, therefore, it would be nice to programmatically capture version info. I just don't think you can. I suspect you'll need to insert that yourself.