Can you put quaker parrots and parakeets in the same cage?

They can. Under certain circumstances.

It would be much easier if they were both chicks or younglings growing up together, as they will see one another as siblings.

Parrotlets are very territorial birds with a big attitude and tend to have "grumpy periods" on some days. A parrotlet can easily break the beak, leg or wing of a budgie. A broken wing can be repaired fine. But a broken leg or beak can make their life too difficult (bird legs are usually too thin to be repaired by a vet).

You can try having the parrotlet and budgie together in the same room under your supervision to see what happens. Keep your eye on the parrotlet for aggressive behaviour. Budgies themselves are not aggressive or territorial, so they will not attack.

If they don't seem to get along, buy separate cages, have one bird in each cage, then place the cages next to each other, forcing the two birds to be neighbours but out of each others reach. Over time they will learn to tolerate each other and may even be witnessed chirping at each other.

Once they seem to get along as neighbours, try again with them together.

If they are housed together, the cage should be relatively tall or wide, so there is plenty of space for them without any arguing over who gets to sit where. There should also be more than one feeding bowl, because remember it is the parrotlets nature to be territorial, they cannot help the way nature made them to be. You can domesticate and train a dog, but at the end of the day it will always still have the instincts of a wild fox/wolf.

Always keep a close eye on them when they are being taught to get along. If the budgie ends up being bullied, remove him/her from the parrotlet. Depressed birds will pull their own feathers out. If fighting takes place, protect the budgie, because it is the weaker and most defenceless one.

They can be taught to get along, but it will take time. It would be much easier if they were chicks growing up together.