Is it true that to produce a blue eyed baby, both parents must have blue eyes?

No-many variables. You get 1 eye color gene from both parents.Brown dominates, so they could be brown with 1 brown & 1 blue gene (Bb) or 2 brown genes (BB).A blue eyed parent would only have blue genes(bb).Two Bb brown eyed parents could conceive blue eyed babies every time but statistically only 25% is expected.A BB brown and bb=blue eyed parent would have all brown eyed babies.Two bb blue eyed parents have blue.
No. Eye color has complex inheritance but brown vs blue is not too difficult. If the father is dark brown due to having both genes brown, then the child must get one brown from his dad and he will have brown eyes. If mom is true blue, she has both genes blue and her child will get only that color gene from her. If dad has one brown and one blue then 1/2 of his children will get blue eyes.