Edit Right now I'm finding parts of conversations in Chrome. It's in the AppData\Local\Google\Chrome\User Data\History Index. Conversations are in clear text without any structure, it contains the name and date, that's all.
This is good news! Chrome can save plaintext versions of web pages in the "c2body" field - it's pretty handy in terms of evidence.
Do you have any experience with SQL? If so, you can export and open that History Index file (using something like
SELECT info.time,
datetime(
pages_content.c0url,
pages_content.c1title,
pages_content.c2body
FROM info,
pages_content
WHERE pages_content.docid = info.rowid
You can then export these results as a CSV (or whatever).