I don't believe that either translation for this term (as referenced by Ana Vozone) is correct. They both refer to "circuit court". According to Marcílio Moreira de Castro:

"Circuit Court. sinônimo de UNITED STATES COURT OF APPEALS" (p. 478).

Under "organização judiciária", Castro explains the differences in court organization between Brazil and the US:

"Nos EUA, existem três instâncias de Justiça Federal:
1st – United States District Court.
2nd – United States Court of Appeals, também chamados de Circuit Courts.
3rd – Supreme Court of the United States
(...)
No Brasil, a Justiça Federal é organizada nesta ordem:
1a instância – Juiz Federal.
2a instância – Tribunal Regional Federal.
3a instância – Superior Tribunal de Justica e
Supreme Tribunal Federal.
(...)

A organização da Justiça Estadual no Brasil e nos EUA:
Nos EUA:
1st – trial court.
2nd – state appellate court.
3rd – state supreme court
(...)
No Brasil:
1ª – juiz de primeira instância.
2ª – Tribunal de Justiça." (p. 286)

In Dicionário de Direito, Economia e Contabilidade, 4. ed., Editora Forense

Therefore: "(federal) circuit court" and "state appellate court" translate as "tribunal de segunda instância" and "Tribunal de Justiça", respectively. These have nothing to do with "Ofício de Justiça"!

To the best of my knowledge, given the different legal structures, there is no such thing as "Office of Justice" in the US as we understand it in Brazil. Nonetheless, that's the translation that's being used by some registry offices in Brazil. Example:

3rd. Office of Justice: https://www.google.com/search?q="3rd office of justice"

Here's a more detailed explanation of its meaning, as quoted from JusBrasil:

A nomenclatura "juízo" tem significado preciso em direito. Corresponde à célula mínima de jurisdição que, no organograma do Estado, permite o exercício do poder jurisdicional. Na justiça comum estadual e federal, o juízo corresponde à vara. A cada juízo corresponde, no mínimo, "um ofício de justiça", que é chefiado pelo escrivão ou chefe de secretaria, ou secretário. O mesmo juízo pode ter mais de uma secretaria ao seu serviço, caso em que se deve identificar cada um dos ofícios ligados ao mesmo juízo. O escrivão é o serventuário público encarregada da chefia dos ofícios de justiça. https://www.jusbrasil.com.br/doutrina/secao/art-150-secao-i-...

It was a long explanation. I hope it helps somehow or at least guide you in the right direction. Cheers!

PS: Wikipedia shows "oficial de justiça" translated as "bailif", and there is also a mention of "bailif's office" in one of their articles; however, I don't believe this would be an appropriate translation for "Ofício de Justiça". It has more to do with enforcement of a court order, court security, etc. rather than registry of birth certificates, which, by the way, are the responsibility of each state. In Arizona, for instance, such registries are managed by the local county Vital Records Offices: https://www.azdhs.gov/licensing/vital-records/index.php#regi...

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Note added at 3 hrs (2024-02-02 00:54:56 GMT)
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Lest I forget, "registry office" translates as "cartório de registro".

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Note added at 6 hrs (2024-02-02 03:51:22 GMT)
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Some remarks on usage:

The term "Office of Justice" has been used on SEC's website. SEC stands for U.S. Securities and Exchange Commission. It appears in a document titled "Descriptive Memorandum of Properties to be incorporated to the Equity of Banco Bradesco S.A." (p. 77). Under "Registry Office" (cartório de registro), it shows "3rd Office of Justice of São João de Meriti" (p. 103). Here's the link: https://www.sec.gov/Archives/edgar/data/1160330/000129281416...

Waze.com shows "Notary 3rd Office of Justice of Duque de Caxias - RJ" and provides a link to their website, named "Cartório do Terceiro Ofício de Duque de Caxias".

Waze: https://www.waze.com/live-map/directions/br/rj/notary-3rd-of...
Website: https://www.3oficiocaxias.com.br/