If you want to apply for migration services from the Swedish Embassy in Belgrade, you would need to prove that you live either in Serbia or in Montenegro. Otherwise you would be referred to the country where you legally reside.
You are hereby referred to the Article 6 of the Visa Code Regulation (EC) No 810/2009 of the European Parliament and of the Council of 13 July 2009 establishing a Community Code on Visas (Visa Code) (europa.eu)
The Embassy of Sweden provides migration services (e.g. submitting biometrics, ordering of residence permit cards, performing interviews, application for Schengen visa or residence permit, etc.) to the applicants who are citizens or residence permit holders in the country where the Embassy of Sweden is situated or is a competent authority for.
Our practice is therefore limited solely to Serbian and Montenegrin citizens, or to the persons with residence permits in these countries, without exceptions. Other inquiries have to be respectfully referred to other authorities.
If you live in Serbia or Montenegro and want to apply for Schengen visa in Belgrade, please send the copy of your Serbian/Montenegrin residence permit by email, and specify details of your and your immediate family members´ stay in Serbia or Montenegro: the grounds for your residence permit in Serbia/Montenegro, economy plan, place of stay, future plans after the intended trip to Sweden, etc. You would then be referred to either make a phone call and book time for your application, or to contact the Embassy of Sweden in the country where you legally reside.