No. An apostille is issued for a document, not for a country: the certificate attests the signature, the capacity of the signatory and the seal on the document, and that attestation carries the same weight in every state where the Convention is in force. The real answer lies not in a second apostille but in the number of originals: if two institutions want the original at the same time, one sheet of paper cannot be in two places.
No destination field on the certificate
The model certificate annexed to the Convention has ten numbered fields: the country of issue, the signatory, their capacity, the seal, the place and date of certification, the certifying authority, the serial number, and the authority’s seal and signature. None of these fields asks where the document is going. The clerk at the valilik or kaymakamlık (the provincial or district governor’s office in Türkiye) still asks about the destination, and the reason can be read in the service standard table of the Çankaya District Governor’s Office: administrative documents going to a contracting state receive an apostille, those going to other countries ordinary legalisation. The question serves to choose the procedure; the answer is not written on the certificate. A document apostilled once can later be used in another contracting state.
One original, two institutions: the real constraint
Whether a foreign institution keeps the original of an apostilled document in its file or returns it is that institution’s own rule. When the same diploma or the same civil registry extract is to be presented to two institutions in two countries in the same period, and both want the original, the way forward is to obtain two separate originals from the issuing body and to have each apostilled separately. On the paper route this means going to the counter with two documents; no fee is charged at the provincial and district governor’s offices. If one of the institutions accepts a certified copy, a notary’s true-copy certification can stand in for the second original, but that copy is a separate notarial act and needs its own apostille. The difference between a photocopy and a certified copy is treated on its own question page on this site.
Two countries may want two different things
The same document does not receive the same treatment in two countries. The German missions in Türkiye accept the multilingual birth, marriage and death extracts without an apostille and without translation; another country may ask for an apostille and a translation on the same form. In that case what is prepared is not two apostilles but two different files: in one the multilingual extract on its own, in the other the apostilled and translated version. Country by country differences are covered in the country guide pages of this site.
Country selection in e-Apostil and the open question
In e-Apostil, the official Turkish electronic apostille system, a country is selected during the application, and the system’s FAQ states that no e-apostille is issued for countries outside the Convention. The electronic document is a file; it can be copied and every copy can be checked by uploading it to the official verification page. Whether one application covers two countries, in other words whether the selected country is written into the certificate, could not be verified in this site’s survey. The service fee is charged per document, so two applications mean two fees; the amount is published on the official page.
Two languages double the translation work
Where the two countries work in two languages, each language gets its own translation, each translation its own notarial certification, and if the receiving institution wants an apostille on the translation too, each notarial certification gets its own apostille, issued by the governor of any province or the district governor of any district. Whether two translations can accompany one original, or each has to travel with an original of its own, is asked of the institutions, and the number of originals is planned from their answers. A translation office builds that plan from the written requirements of the institutions receiving the document and makes no assumptions before the institutions have been asked.