As a translation, "relief proceedings" is the term approved by the Suprema Corte de Justicia de la Nación in its publication "Relevant Decisions of the Mexican Supreme Court" (2006). You'll have to decide for yourself how qualified that court is to render this term in English though.

Anyway, as Helena and Maria have said, most translators leave it as Amparo proceedings or writ of Amparo. Javier Becerra (who IMO is qualified) also gives us "constitutional remedy" and "constitutional relief under a writ of Amparo" which nails it really (and is probably a good term to use the first time it appears in any document where this term would be unfamiliar, i.e., not specifically part of a court case).