Can a brown hair brown eyed mother and a brown hair green eyed father produce a blue eyed child?

Well it comes down to Gene's at the end of the day, and not just the parents which conceived the child. Brown colored eyes are the most dominant gene. So usually these take over from any blue or green.

Blue colored eyes have the weakest gene, so unless your families are prone to blue eyes your child will have a less of a percentage of getting blue eyes. But, there is no saying that you cant produce these. A friend of mine comes from a family of 5 siblings and mum and dad, all her siblings including mum and dad where all fair haired & skinned with really pale blue eyes.

However, my friend had olive skin, really dark eyes and dark hair.... turned out one of her late ancestors where of foreign blood. My father has pale blue eyes and my mother has hazel eyes. My two sisters both have Deeper blue eyes than my Father (sort of a mixture of both mother and fathers eyes) but I have bright green eyes.