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A U.S. citizen or green card holder living in Canada has to report their worldwide income to the Canada Revenue Agency (CRA) and to the U.S. Internal Revenue Service (IRS). If done properly, a taxpayer most often does not owe any taxes to the IRS thanks to the Canada-US Tax Treaty. This treaty allows a foreign taxpayer to use foreign tax credits against U.S. tax for the amount paid in Canada.
The US will give you a Foreign Tax Credit for the Taxes paid to Canada on income you earned there - but you may still owe some to the US, if you have other unearned income such as interest and dividends.
If you did make US quarterly tax payments, you would get a refund for the Canadian taxes paid on your Canadian income. You do not have to pay estimated taxes to the US. Here is an awesome link for more detailed information: The USA/Canada Tax Treaty Explained
