Shooting and killing pigeons is against the law in some states but other states may allow you to shoot pigeons.
For example in New York City where you usually see a lot of pigeons it's against the law to shoot those pigeons but mostly because it's also against the law to discharge a firearm in the city limits.
So really you'd be getting charged for discharging the firearm in the city limits and likely not charged specifically for shooting the pigeon.
If you were out in the country far away from neighbors or anyone then you could likely shoot the pigeon and kill the pigeon even if it were against the law.
Because then nobody else would really know about it unless they were watching you.
Individual states and jurisdictions can have there own laws so in one place it can be illegal to shoot pigeons and other places may allow the shooting of pigeons.