The database QC connects to for initial startup is controlled by the file siteadmin.xml that is located in the file C:Program FilesHPQuality Centerjbossserverdefaultdeploy20qcbin.war with QC 10. (It was in a different file in QC 9.) You can edit 20qcbin.war using a tool like 7zip in place or you can use other tools that will extract the WAR (much like a zip) and then edit the file and then recreate it.
You should make a backup of this WAR file in case you corrupt it by accident.
The full path to siteadmin.xml is : C:Program FilesHPQuality Centerjbossserverdefaultdeploy20qcbin.warWEB-INFsiteadmin.xml. I use notepad to edit the file with wordwrap off. Look for the DBName setting.
Keep in mind there is another set of steps to get all your projects working with the new server, but this should let you login --if-- when your DBA's moved you to a new server they correctly setup userid's etc.