The answer is no, two homozygous, brown-eyed parents could not have a blue-eyed child. The reason is very simple. If the brown-eyed trait is dominant, that means that you only need to have one of the alleles to code for a brown-eye phenotype. Thus, a person with either the BB or Bb genotype will have the brown-eyed phenotype. The only way for a person to have the blue-eyed phenotype is to be homozygous recessive with a genotype of bb.
Now the question states that both parents are homozygous (meaning they have two of the same alleles for the trait), and that they are homozygous for the brown-eyed trait. This means that both the mother and the father’s genotype is BB. You may already be able to see why the answer is no, even without a Punnett square. If both parents have the BB genotype, that means that there is no little “b” allele—which codes for blue eyes—to pass on to the next generation. However, we can chart this by using a Punnett square.
BB x BB = ¼ probability of getting either BB, BB, BB, or BB.
Thus, you can see that the only allele being passed down is the big “B” allele, which codes for brown eyes. There is thus a 100% chance that the child will have at lease one big “B” allele (and, in fact, a 100% chance that they will actually have two, and thus have the BB genotype). Because brown eyes is a dominant trait, you only need one big “B” allele to display it. It is therefore impossible for the child to have blue eyes.