Welcome to Android Central! Are you trying to back up all contacts that are on your Note 2 to your Google account? Or are you trying to get all contacts from all accounts displayed on your Note 2?
What account are the non-Google contacts associated with? Is it the "Phone" account? The problem with entering contacts to the Phone account is that it doesn't sync directly with your Google account--you can only sync those contacts using some kind of 3rd party software (typically provided by the wireless provider, like Verizon's Backup Assistant Plus).
In order to get those Phone account contacts to your Google account, you need to export them first to a .csv file, then import that file into your Google Account. The following are general steps, but the details might be a little different for you (sorry, but I don't have a Samsung device):
Go to your Contacts/People app, and select only your Phone account to be displayed.
Tap Menu, then find the Export option. Export the contacts to a .csv file (to your SD card, not the SIM).
Now you can import that .csv file two different ways--directly on your phone, or on your computer. On your phone, in the Contacts/People app, open the Menu and find the Import option. Import that .csv file, and make sure you import it into your Google account.
Alternatively, you could transfer the .csv file to your computer (either directly by unmounting the SD card, removing it, and plugging it into your computer; or uploading it to a cloud storage like Google Drive and then downloading it to your computer). Then on your computer, go to Gmail on your browser, click on the dropdown at the upper left that says Gmail, and select Contacts, click the More button and then Import, and then select the .csv file.