The short answer

Yes — a small piece of walnut is a fine, healthy-fat treat.

Why

Walnuts offer healthy fats and are safe for budgies, but they're rich, so only a little. Use fresh, unsalted nuts and discard any that smell off, since mould is dangerous.

How to feed it

Offer a small crumb of shelled, unsalted walnut.

How much & how often

A small piece occasionally.