Ok, it sounds like the iPad is asking you for your own Passcode, not a verification code.
The Passcode is not a code that can be sent to you because that code only exists on the device itself... but it must be one that was set at some point since purchase and has now been triggered as part of the update.
I'm afraid we are at Apple's mercy here as there is only one way to get past this point if you've forgotten that code, and that is to reset and restore the iPad.
The following procedure is a special device firmware update mode. It will erase your Apple device, remove the passcode, and then re-sync user data once it is done (provided you have a backup of some kind).
Make sure you have downloaded and installed the latest version of iTunes on your computer before attempting this:
(Mac users can run System Preferences > Software Update (or “Apple Store” for OS X 10.9 users) > Check now to get the latest version for their system in most cases.)
-> Open iTunes on your computer and connect the Apple device directly to it
-> Turn off the Apple device (press the sleep/wake button on the top until you see the red slider and slide the red slider across to the right OR until the device completely shuts off).
-> Hold down the sleep/wake button (on the top) AND the home button (on the front) for 10 seconds
-> Release the sleep/wake button but CONTINUE to hold the home button, until...
-> iTunes should alert you it’s detected a device in recovery mode, click OK and continue along for the restore.
Note: it may ask whether you wish to restore a backup from iTunes or from iCloud. You should use which ever has the latest backup of your personal data.
If it doesn’t work the 1st time, start from the beginning and try again. If iTunes *still* doesn't recognize the device, plug it into a DIFFERENT USB port and try once more.
If you receive misc errors when attempting to install/uninstall or sync after installation, try temporarily disabling any anti-virus or firewall software you may be running.
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Thanks and good luck,