Do we know where the OP is located? Lol, good points being made by all, but we lost the OP in the process...I hope that the OP realizes that we here are very passionate about our birds, and birds in-general, and in the US, the UK, Canada, etc., it's unfortunately very common for people to "hoard" parrots or buy a parrot and not house/feed them correctly when they have every opportunity to do so, they just don't...Sometimes we forget that different countries have cultural, environmental, and social differences and standards than we do when it comes to keeping/breeding birds, and we have to be reminded of this from time to time...The one thing I can assure the OP is that everyone here means well and gives pretty good information, and for the most-part are extremely educated in owning, raising, and many also in breeding parrots, so we can answer your questions...
I was confused as to what the OP's main goal was, I wasn't sure if he wanted to breed this pair or tame this pair as pets, or both I guess...Doing both is extremely difficult, as once a pair of birds of the same species and opposite sex are bonded-closely with one another, and also living in a small "breeder cage" with a nest-box attached, they are going to have little to no interest in having any contact with humans at all...There are extremely rare exceptions to this, but they are very few and far between, and usually the circumstances surrounding these situations are quite unusual...
For the most part, when you have a breeding-pair of parrots like you do that are already bonded-closely and living together in a cage with a nest-box, and have already been mating/breeding, it's extremely difficult to hand-tame them and have either one of them bond with you again, unless you separate the pair into their own cages and into different rooms so they cannot see each other or have any contact with each other anymore...And even that doesn't always work if they are still in the same house and they can still hear each other or sense that their mate is still nearby...A lot of the time when people attempt to separate an already-bonded breeding pair of parrots, what they end-up with is two parrots living alone and doing nothing but screaming all day long for each other...
So usually you unfortunately need to choose one or the other, either keep them as a breeding-pair of birds and work on making their environment and diet better to not only encourage breeding, but to also make fertile-eggs that will survive/hatch successfully more probable, OR you could try to start working with them each individually, away from each other and alone in a different room with just you and one of the birds with a shut-door, and see if you could possibly succeed in taming them, but if I'm being honest with you, usually all that ends-up happening if you attempt this is you getting very frustrated because the hand-taming/bonding process is not working at all, and in the meantime you doing this with them each every single day, which you would have to in-order to even have a chance at taming them, and also the chance of successfully breeding them is also slim to none because of the stress and separation you cause the pair...
So if I were you, if you do really want to breed this pair of Alex's successfully (and maybe then hand-tame one of their babies as a pet-only-bird for yourself), I would do two things if they are at all possible for you to do, in-order to make their environment and their health more apt to successful breeding...#1) I would work on their regular, daily diets...I realize that you may not have access to a commercial pellet diet for you birds, and you may need to feed them a seed-mix as their main, staple diet. There's nothing wrong with feeding your birds (and this goes for ALL of your parrots) a main,
"staple" diet of a seed-mix, AS LONG as it is a seed-mix that is as low in fat as possible, and AS LONG as it is supplemented each day with lots of fresh veggies, fresh dark, leafy greens, and a little bit of fresh fruit (fresh fruit should only be given 2-3 days of the week in small portions due to the high sugar content, especially since they are eating seed as their staple diet)...You do not want to feed them a seed-mix every day that contains any type of Sunflower Seeds, as they are very high in fat, nor should it contain any type of Nuts, especially Peanuts, nor should it contain any type of dried Corn kernels or dried cracked-Corn. If you can either buy or mix/make a seed-mix that contains no sunflower seeds, no nuts, and no dried corn, but instead contains a large variety of healthier seeds that are lower in fat than the sunflower seeds, and that also contains different fresh grains, legumes, etc., this is going to improve their health tremendously, as will giving them whatever fresh veggies and dark, leafy greens you have available to you...The only veggies or greens you can't feed your birds are Onions, Leeks, Chives, etc., any veggie in the "onion family", as they are toxic to birds, and also no "watery" lettuces, like Iceberg Lettuce, as they contain little nutritional value at all and are mostly water...And never any Avocado for a fruit, it is also toxic...
#2) Trying your best to find them a larger cage to live in...The cage that you have them in is incredibly small for one Alex, let alone two Alex's that you want to breed and raise healthy babies...No one is telling you this to be critical of you or mean to you, but rather to educate you for the good of your birds. In that small of a cage your birds are getting no exercise at all, and on a seed-mix diet with little to no exercise they are quickly developing Fatty Liver Disease, and also possibly Diabetes...And a HUGE ISSUE that may play a part in successfully breeding them is that Alex's are extremely prone to obesity...So the combination of the all-seed-diet that is very fatty and then living in such a small cage where they are getting no exercise at all and can hardly even move around in or flap their wings in is likely going to result in the birds becoming overweight, and eventually obese, which can and will interfere with the female successfully producing healthy eggs, being able to lay them properly, and the eggs themselves being healthy and able to support an embryo. [B/] As a parrot gains weight, their Keel Bone (the long, pointy bone that runs down the middle of their chest and belly) is pushed further and further out and away from their abdomen because of the fat accumulating in and around their bellies...This is called "Sternal-Lift". And when a female bird is overweight and develops Sternal-Lift, what basically happens is the location of their internal organs changes, and this includes their Reproductive Organs, specifically the Oviduct where the eggs sit once they are made but before they are laid...And what happens with overweight/obese female birds is that their bodies make the eggs, but instead of the eggs being deposited in the Oviduct and then laid by the female, they end-up being deposited outside of the Oviduct and are just dropped into the free space in their abdomen...And this is the one way the Egg-Binding occurs that is 100% fatal for the female without surgery to remove the egg...It can never be fertilized by the male and it can never be laid by the female, it has nowhere to go, it just sits inside of their bellies and basically rots, until the rotten egg causes a systemic bacterial infection in the female...And without surgery to remove the egg from the female's belly, the female dies a long and painful death...And obesity is the #1 cause of this type of Egg-Binding, and is most often seen in species of Amazons and Alexandrines due to their propensity for obesity. The other type of Egg-Binding is where the egg is made and IS deposited into the Oviduct, but due to nutritional deficiencies that the female has because of their daily diet, the eggs are very soft or underdeveloped, and this makes them impossible for the female to expel from their body...The best way to avoid this type of Egg-Binding is by always providing a Cuttlebone and a Mineral Block to your female so that she gets extra Calcium and Phosphorous, but also by providing them a daily diet that is full of fresh veggies and dark, leafy greens, and by eliminating the sunflower seeds, nuts, and dried corn from their diets...
However, if your birds are living in a cage large enough for them to move around, climb up and down the bars, and most importantly be able to flap their wings without them touching the cage bars, this will help tremendously because they will be getting exercise to keep the fat from accumulating in their bodies...The other issue with a cage that is far too small is that it can also cause them a lot of stress every day, because they cannot ever get away from each other, they can't ever really move around, they can't open their wings, etc., and this can cause them a great amount of stress, which is obviously not good if you're wanting them to successfully breed and lay fertile, healthy eggs that will survive and hatch...So if there is any way at all that you can find a much larger cage to keep this breeding-pair in (with the nest-box attached to the outside of the cage through a door, so it's not taking up a bunch of the cage space that they can use), their overall physical and mental health is going to improve greatly, and this will improve the odds of them breeding successfully.
