The supplements that may be necessary are calcium and vitamin D for all female birds that are laying eggs or may be laying eggs in the future whether you intend them to or not. For birds that are stubborn seed junkies, more complete supplements may be necessary. Of course as Sailboat says a balanced complete diet is the ideal way to get nutrients into their diets but some birds just ignore that memo. I wouldn't put any supplements in their water, however, unless the supplement is sugar free because many birds like to bathe in their water bowls. If your birds drink out of tube style water dispensers dish bathing wouldn't be a problem unless they try to bathe or splash themselves under the drinking tube somehow. I know that many top breeders of champion show budgies use supplements in their water but most use tube waterers. You really need to wash the dispensers thoroughly and frequently to prevent bacterial growth but many dispenser users don't wash them that often.
I can't see why an avian vet would recommend supplements in a healthy bird, especially a male bird, that eats quality pellets, veggies, and fruit. I thought quailty pellet manufacturers recommended against using additional supplements because it would like giving the supplements twice and it's possible to give too much.