I honestly wouldn't recommend housing conures and budgies together because a conure could seriously injure or kill a budgie. There are always exceptions, but I wouldn't recommend it in general.
That said, I've had budgies who honestly didn't care or one that *wanted* to be around a conure without physically interacting with him. He was one pissed off bird having to chase a little blue pest around his cage! Of course, I rehomed her (the budgie) for her own safety....
Smurf hiding away in Noel's cage, taking a break from his girls. They pretty much ignored each other. (excuse the messy looking cage!)
Charlie and Normie. Normie (short for Normand, now[?] Indie) had an unhealthy fascination to being in Charlie's cage and would squeeze through his cage bars to go play with his toys, eat his food, drink his water, etc. Charlie was not in the least bit pleased and would chase her around the cage... only, she could easily escape him via flight while he was reduced to climbing and walking (although the cage is pretty large, still not large enough for him to fly in!)
I no longer keep budgies, as cute as they are.