No - lycanthropy can only be passed on by a werewolf bite.
In “Fantastic Beasts and Where to Find Them”, it’s stated that humans will only become werewolves when bitten.
“Humans turn into werewolves only when bitten. There is no known cure, though recent developments in potion-making have to a great extent alleviated the worst symptoms.”
- Fantastic Beasts and Where to Find Them
This seems fairly conclusive - the condition of lycanthropy can only be passed on through a werewolf bite, so Teddy Lupin would therefore not “inherit” being a werewolf.
In addition, a Pottermore writing by JKR says it’s never seen to be hereditary.
In a writing on Pottermore, JKR specifically addresses whether lycanthropy can be inherited from a werewolf parent, and says there’s never been any sign of this happening.
Werewolves generally reproduce by attacking non-werewolves. The stigma surrounding werewolves has been so extreme for centuries that very few have married and had children. However, where werewolves have married human partners, there has been no sign of their lycanthropy being passed to their offspring.
Since Teddy Lupin is the only child of a werewolf that we know of in the series, this seems like an intentional way of implying that he wasn’t a werewolf.