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Unfortunately, you cannot add a country code field to the phone number quick tool.
However, you can use regular fields to capture this information. Something like in this form: http://form.jotformpro.com/form/50826764626968 in which I used 3 number fields and custom CSS to arrange them in a similar way.
If this could work for you, feel free to clone the form to your account to check it further.
Let us know if you need further assistance.