We have several budgerigars and from time to time my partner sees an egg in their cage and disposes of it immediately. They are possibly fertile because we have both male and female budgies however the cage doesn't have a nesting area so it's probably unlikely the female would be able to keep the egg warm enough for it to hatch.
We keep a few small tree branches / leaves and the odd piece of fruit / veg on the bottom for them to nibble on but it doesn't really cover the entire area. Most of the eggs seem to be laid down the bottom directly on the bottom of the cage. While well sheltered it's outdoors so there would be cool air passing under the eggs.
Because budgies are somewhat skittish it's hard to know how long (if at all) they are trying to nest on them. I wondered if removing the eggs quickly may cause some sort of anxiety or if maybe it's better to leave them in place a while until they "give up" on them trying to hatch?
We're not especially interested in breeding them but if that seems like the most humane thing we're not certainly not opposed to it and I'm sure we can find them loving homes. If that sounds like a better option I'd also be interested to know at what point eggs that haven't hatched should be removed?