If you put live plants in a bird's cage, they will probably (like, almost certainly) be eaten or shredded. Which is fine, if that's what you're planning, but you do have to plan for it.
I found this page that lists some safe and toxic plants to work with.
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One plant that I can say for sure that is safe and that birds love to play and forage in is wheatgrass. You can plant the wheatgrass seeds and then scatter treats among the blades of grass once they sprout up and it's a fantastic and fun foraging activity. But birds make short work of it, so it's kind of like... you spend a week or whatever growing the tray of grass for one day of activity LOL!
Crimson bellied conure- Tequila
Greencheek conure- Sierra
Pearled cockatiel- Malibu
Cockatiel- Volkan
Yellow budgies- Pina Colada and Houdini
Blue budgie- Lightning
White and blue budgie- Ciroc
It's a brilliant idea and your birds will love it, but it definitely won't last and it will be very messy. I've had live plants in my cages before and I stopped because of the mess and the upkeep of it, but the birds do love it. Make sure you use parrot safe plants, and that they can't get to the soil (they like to eat it which could be dangerous depending what's in it).