How do USA Estimated Tax Payments Work?

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If you are still a US citizen but living elsewhere and filing returns, but not owing tax perhaps because substantially all your income is excluded by the Foreign Earned Income Exclusion, then you are in one of the safe harbors for estimated tax. Specifically, if you reported zero tax liability for the prior year (2020) -- this is line 24 on 1040, with adjustments in some relatively rare cases -- and that return was for a full year, your minimum required in-year payment for 2021 is zero, nothing, nada, nil.

The usual alternatives, to which you may have seen reference, are that to avoid the underpayment penalty you must pay (by withholding, timely estimated payments, or any combination) either 90% of this year's tax (which at present you don't know, and you may be able to estimate well or poorly) or 100% of last year's tax (which you do or soon will know) on a full-year return as above, increased to 110% if last year's AGI was over $150k. See Form 1040-ES and its attached instructions which you can download. If you underpay, the penalty is proportional to the amount and time duration of the underpayment, and thus indirectly to either this year's tax or last year's tax whichever gives the lesser payment requirement.

You can of course still make a payment(s?) if you want -- you will need to pay not later than next April (to avoid interest, although filing from out of country you don't get the additional failure-to-pay penalty until June), and if you don't have a better use for the money in the meantime, there's no harm in paying now.

Be advised in the US tax context 'expatriation' means something quite different from just living elsewhere: if you renounced your citizenship (or had naturalization revoked), or in some cases relinquished, abandoned or had revoked your "green-card" (lawful permanent resident) status, with a net worth of at least $2 million or income averaging for the previous 5 years at least an inflation-adjusted amount currently $171,000. Such a person is subject to a special tax described here and in more detail in pub 519 but NOT to the normal 'citizen' or 'resident' income tax, since after expatriating you are neither of those.

Also, remember FBARs are required for accounts over $10k regardless of any tax or tax filing