Yes — Szilvia is Hungarian
Reliable name references identify Szilvia as the Hungarian form of Silvia. It is a feminine given name used in Hungary.
The “Sz” spelling is a clue
Hungarian uses the digraph sz for the /s/ sound. An English reader may see “sz” and expect a z-like sound, but a Hungarian reader sees a normal spelling pattern. That makes Szilvia visually identifiable as Hungarian in a way that Silvia is not.
Is the name originally Hungarian?
Not in the sense of having been created from a native Hungarian word. Szilvia is the Hungarian form of the older Silvia name family, which has Latin and ancient Roman roots. Many established names enter a language from older traditions and then develop language-specific forms.
How is it pronounced in Hungarian?
Approximately SEEL-vee-yaw, with first-syllable stress. The exact IPA given by Behind the Name is /ˈsil.viʲ.ɒ/.
What is Szilvia’s Hungarian name day?
Behind the Name lists November 13 for Hungary. Name days, or névnapok, are a familiar part of Hungarian naming culture.