Seemingly no matter what options I choose on the Volunteer Report it throws the error "DB Error: unknown error" and I don't see anything in the Apache error logs.
What's the best way to troubleshoot this and see the actual error? Thanks.
You are looking in the wrong place.
Check your PHP logs on your webserver.
Check the civicrm logs in /civicrm/ConfigAndLog or wherever you have configured that folder.