They do! I started off with just one budgie and she was lonely so I rescued another. I didn't have a preference for male or female but I ended up with a male and a female and she was very broody and kept laying eggs even before I got the male. A friend kept asking me if he could have any babies if I was going to breed them and since she was extremely broody I decided to let them breed. They had four babies and I gave two to my friend (2 males which was lucky because they're his first birds) and I kept the two females. This was almost two years ago and they've shared the cage the entire time.
I think it's because they're mother and daughters that they get along. I was worried the mum might be a terrible mum since she came from awful conditions and had a lot of problems but she's an amazing budgie, and the dad is so sweet. I got him from a rescue and the woman had loads of female budgies and they all hated him for some reason but it was love at first sight when he met my budgie. He dotes on the three girls and is always feeding them and singing to them. The females all feed each other sometimes too so it's not just a mating thing with them. Even the siblings fed each other when they were still growing up! I do have to make sure I have three bowls in the cage otherwise they fight over food as the girls like their own bowl, and the parents share one. I'm quite lucky since they don't show many hormonal behaviours and don't seem interested in having more children. The two daughters have never laid an egg and don't seem too fussed abut finding a mate and having children either. If I got more budgies I think they would though.
I know you're worried about Rocky having babies and being hormonal. It's a really tough decision to let them have friends of the opposite sex in case they breed. I'm hoping that because she's had some babies and her diet is now good (she was half starved and on a seed only diet when I got her), mine won't want any more. I think the better diet and abundance of food she suddenly got when I rescued her made her start laying eggs but I didn't know that back then. And I'm hoping that the younger ones will never want any.
