Div for the nested quote definitely won't work, the only way to get the second level quote indented is to also nest it within the blockquote. Multi-level indents are definitely supported in mobi, it's just you have no precise controls on the left margin, and zero controls on the right margin.
Check the second pic in this blog post, using multiple nested blockquotes:
Not sure why the first level didn't get indented though, perhaps you're suspicion about <p> tags is correct, you could try removing those. The only other possibility is that the conversion engine may have been changed recently, Kovid has implemented a number of changes in epub to mobi conversion due to a large number of users complaining about everything being indented based on the old logic of converting css margins to mobi blockquotes.
You might be able to get a better idea if this is some issue with Calibre if you convert your problematic mobi back to epub with a debug directory configured, then look in the input directory and there is a file called 'debug-raw.html'. Check this file and see what the code around your problem quotes is. You might want to post that section of code from the original epub and from the resultant mobi to epub debug back here, one of us may be able to provide some more input then.