The short answer

Only a tiny amount of hard cheese, rarely — birds don't handle dairy well.

Why

Budgies are lactose-intolerant, so dairy can cause digestive upset. Hard cheeses like cheddar are much lower in lactose, so a tiny crumb very occasionally usually causes no harm — but it's not something your budgie needs, and soft or creamy cheeses should be skipped.

How to feed it

If you offer it at all, give a crumb of hard cheese very occasionally, and avoid soft, processed or flavoured cheeses.

How much & how often

Rarely, and only a crumb.