The short answer

Yes to the flesh, but never the pit — cherry stones are toxic.

Why

Cherry flesh is a fine sweet treat, but the pit (stone) contains cyanide compounds, exactly like apple seeds and other stone-fruit pits. A budgie must never have access to the pit.

How to feed it

Remove the stem and pit completely, then offer a small piece of the flesh.

How much & how often

A small piece occasionally.