Yes — watermelon is hydrating and completely safe for parrots, seeds included.
What it means for budgies
There's nothing specific to a budgie that changes the verdict on watermelon — it's treated the same as for parrots generally. A budgie portion is tiny — a single bite or a little grated amount is plenty.
The full picture on watermelon
Watermelon is over 90% water, making it a perfect hot-weather food, and unlike apple seeds, watermelon seeds contain no cyanogenic compounds — many parrots enjoy the seeds most of all. The rind is safe too if well washed, though most birds ignore it.
Serve chunks with or without seeds — both are safe.
Chill it in summer; offer on a skewer or in a bowl.