My first guess would be to look at the view "_datasources" in the Tableau Server PostgreSQL db. Are you familiar with getting to that?
Yes I am familiar with _datasources view in postgres but I don't believe it has query info/syntax in it.
I don't know how much investigation you've been doing but here's what I tried.
I searched for *.tde on my server and the files returned were of no use. From what Russell C. has mentioned it looks like the actual data and dictionary info. Query info isn't in it (correct me if I'm wrong).
Next, I searched for *.twb and came across this interesting directory:
In it are files and directories. What you could do is run an extract for the particular data source you're after and once it's done go to this directory and look for entries with a Date Modified date/time stamp of the execution. Once you have that you can open the .twb and/or .tds files in a text editor (I like using the GNU app. NotePad++). Do a search for "select" and Bob's yer uncle you got it.