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If you can't copy your contacts directly to your Google account, you can export them to a file and then import that file to your Google account. You may have to tap the Settings option first. This will set the contacts app to only display contacts stored on your device, which are the contacts that need to be backed up. This will set the contact file to save in your phone's storage. If you're given the option, tap the contacts you want to export. Since you restricted the view to the contacts just stored on your device, you can usually just tap "Select All." You'll see a notification appear at the top of the screen when the contacts have finished exporting. This will ensure that the imported contacts are added directly to your Google account. Tap the file that you just created when prompted. This will import the contacts from that file into your Google account, backing them up online. Sign in with the same account you just imported the contacts to. Look for the contacts from your phone that you just imported. If you see them, they have been safely backed up to your Google account. It may take a little while for the contacts to sync.
Tap the Contacts app on your device. Tap the or More button. Tap Contacts to Display or Display Options. Tap the Phone option. Tap the or More button again. Tap Settings or Manage Contacts. Tap the Import/Export or Back up option. Tap Export. Tap your device's storage. Tap the contacts you want to export. Wait while your contacts export. Tap the or More button in the Contacts app. Tap Settings or Manage Contacts. Tap the Import/Export option. Tap Import. Tap your Google account. Tap your contact file. Visit contacts.google.com in your browser. Sign in with your Google account. Find your newly-imported contacts.