True
In this scenario, the father has the genotype "BbGG" for two genes. He can pass on one allele from each gene to his child. He has the option to pass on either "B" or "b" from the first gene (indicated by "Bb"), and since he is "GG" for the second gene, he will pass on "G". Therefore, he can indeed pass on "B" from the first gene and "G" from the second gene, or "b" from the first gene and "G" from the second gene. However, he cannot pass on "g" because he does not have any lowercase "g" alleles in his genotype.
So, the statement specifically about passing on "B" and "g" would be False.