I wanted to know whether, if I were to marry someone with blue or green eyes with myself being brown-eyed, I could have children with non-brown eyes?

Maybe,maybe not . This depends on you. Brown eyes are dominant and occur if only 1 or both of a gene pair is present. If you only have 1, you could have dominant brown or green or blue if you had a matching recessive gene. If both your eye genes are brown, you can only have brown for your natural kids.