I would NOT cage two Indian Ring Necks together. If you had a large walk in aviary they can live together, however this is not the same as in a cage. They will not have room to get away from conflict. The two birds could end up getting along fine for out of the cage play together. This doesn't mean they can share a cage.
Explain that if she wants a green and a blue IRN they must have their own cage. The cages can be in the same room, the birds can spend lots and lots of time together. It isn't because we want them to be lonely, separate cages are for their own safety. Birds have little fights sometimes, even best friends, in a cage fight can escalate to fatal conflicts. In a large aviary the birds can fly away from the issue and will typically resolve their disagreement peacefully.
Imagine if you were mad at someone, but could never get away to cool off or resolve whatever is bugging you, it would be very stressful! It is much nicer for humans as well as parrots to have a little bit of their own space. IRN are much more independent, compared to species like your adorable conures.
An additional thing, getting a parrot, let alone two is a MASSIVE life changing commitment. Do your best to really understand what you are getting into. You mother also needs to be involved in all of the research. Parrots are for life!