Budgie/Conure= NO, not going to happen, but they shouldn't be allowed to mate AT ALL as Noodles already mentioned, as it's just going to keep them both in breeding-season, which will eventually result in the female of the pair becoming a chronic, continual egg-layer, and both of them can become aggressive, become biters, even become violent...Same thing with the Hammock and the "Hut", they have to go, anything that is creating a small, dark place they can get into/underneath is a huge no-no, including furniture, pillows, blankets, etc...And the "Huts" that you buy in pet shops have killed thousands of pet parrots, 50%+ being different species of Conures, because the material they are made out of is not digestible by birds, and it builds-up in their GI Tract over time, and eventually they are found dead in the bottom of their cage due to bowel-obstructions, crop-obstructions, etc., OR strings that are contained inside of the Huts come loose and hang birds, or hang them from their legs/feet and they actually chew-off their legs/feet to free themselves...No joke, they are bad news...Your birds, none of them, they don't need ANY TYPE of bed, nest, cave, or Box in their cages at all, nor outside in your house anywhere, nor any "Nesting Materials" in the bottom of their cages like bedding, wood-chips, corn-cob bedding, crushed walnut-shells, etc. Just sheets of newspaper or butcher's paper, or paper cage-liners that are in the bottom-tray UNDER THE GRATE so they can't get them and shred them...
Amazon and Sun Conure= MAYBE...Depends on the sub-species of Amazon, and they typically do not live long and are born tragically with deformities, so it's best to do the same thing with the Eggs as soon as they are laid as you need to do with the Budgie/Conure Eggs, and that is immediately remove each Egg when you first see it, boil it in a pot of water on the stove for 20 minutes, let it cool to room-temperature, mark it with a pen so you know it's already been boiled, then put it right back at the bottom of her cage so she can lay on it...And then do the exact same thing with each additional egg she lays, and then once she is done laying the ENTIRE CLUTCH of Eggs and they've all been boiled and marked, you need to just let her lay on them for as long as she wants to. As soon as she realizes they aren't going to hatch she'll lose interest and stop laying on them, and once you've seen her stop laying on them for a good couple of days, you can THEN remove them all and throw them out...This should knock her out of Breeding-Season and should stop her laying continual Egg clutches, which is what will happen if you just remove an egg and throw it out...
***I would do your best to stop any of your birds from mating, not so much because they are different species or because they will produce fertile eggs (99% they won't be fertile eggs),, but rather because the reason the female is laying eggs to begin with is because of the constant mating. 8 times mating in one morning is going to result in the female laying infertile eggs, you can almost bet on it, and that's not good because with every single egg your female bird(s) lay, they are at risk for becoming Egg-Bound, which is 100% fatal to them without immediate Veterinary intervention...And that's in-addition to the negative-behaviors that all of them who are mating/hormonal will start displaying, which is going to likely cause issues between you and your birds and your relationship with your birds, as well as between them....So I would not let them out of their individual cages together unless you are supervising them the entire time, and if/when they start to get close and look like they are going to mate, you need to separate them, and keep doing that until it stops when they are both knocked-out of Breeding-Season. As long as they are allowed to mate, feed each other/regurgitate for each other, etc., they will be in Breeding-Season. They can have their cages next to each other so they can talk to each other and see each other, but as far as contact with each other it's just not a good idea at all, and usually is very harmful to the health of the female. They can become Nutrient-Deficient/Malnourished regardless of their diet, they often become Anemic and Hypoxic (their Beaks, Ceres, and their Skin actually turns a dark bluish-black color), and of course they have a 50/50 chance of Egg-Binding with each egg...So it's not something you should think about as being "mean" or "cruel" for you to do, but rather the responsible thing that you have to do as a bird-owner because you care about their well-being...
