The logs are indicating that your AppURL is incorrect:
20130628.20:14:56 UTC : QBWebConnector.SOAPWebService.updateWS() : Actual error received from web service for clientVersion call: <The request failed with the error message:
<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.0//EN">
<html><head>
<title>301 Moved Permanently</title>
</head><body>
<h1>Moved Permanently</h1>
The Web Connector is hitting your server, and your server is returning a "301 Moved Permanently" response instead of a "200 OK" response like it should be.
Are you 100% positive that your AppURL is correct?
Can you actually post your AppURL, so that we can verify that it's correct?