it very well might, but it wouldnt affect natos ukraine effort at all. they'd throw a load of sanctions at serbia and call it a day, and since serbia unlike russia does not have any economic leverage against europe, and is wholly surrounded by nato, that would just be a complete death sentence for serbian economy. serbia also, unlike russia, is not in a position to be self-sufficient, they'd collapse really hard if this were to happen.
basically my point is that russia is strong and independent enough that it can afford to challenge nato like it did. it will probably suffer in the long term as a result, but it can do it. serbia is not in a position like that, it is landlocked and surrounded and cut off from any potential outside ally. it would be complete suicide to lash out now, and i think they know it well enough to not do it.
what might happen though is republika srpska trying to separate from bosnia without serbia itself getting openly and actively involved, i could see that. but without serbian support this would just be a bosnian interior dispute, at most a civil war inside bosnia, even less of a concern to nato than the bigger scenario where serbia does get involved.