Translation:Žofie was already married on her eighteenth birthday.
Yes, but it would sound very very strange. To say she got married we would rather say:
"Žofie se o svých osmnáctých narozeninách vdala/provdala."
Seems correct, but notice the huge difference in the frequency https://books.google.com/ngrams/graph?content=already+married%2C+already+wed&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Calready%20married%3B%2Cc0%3B.t1%3B%2Calready%20wed%3B%2Cc0
I will add wed as I found some examples that were relevant enough. But I will not go around adding it everywhere, we have much bigger problems than this very rare thing.
Na is not really used about times or date with locative. With accusative "na své narozeniny" would be possible. O + locative is pretty idiomatic for o narozeninách, o prázdninách, o Vánocích, o Velikonocích and similar.