There is no API for Wiktionary at the dictionary level, only at the wiki level, which is probably not much help. See this question for more details: How to retrieve Wiktionary word content?
So there is no API that will allow you to query for synonyms or even to query for Georgian language entries or data.
You can fetch Wiktionary pages either in raw wikitext or in HTML, and you can download entire database dumps in an XML format, which you can try to parse from scratch.
One thing that will help in the case of Georgian is that is has a unique alphabet so the vast majority of Wiktionary entries in the Georgian script will be for the Georgian language. (But there are also a very small number of entries for words in Laz, Mingrelian, Old Georgian, and Svan which will also use the Georgian script.)