If parents are faithful to raise their children according to the Bible, does God promise to eventually bring those children to know him?

I don't think it promises that absolutely.

"Bring up a child in the way he should go, and when he is old he will not depart from it," is the Bible verse that is quoted to suggest that if you do your job right as a parent they will believe.

Let me make it complicated.

There is a translation problem in that verse. It may mean what I just quoted, but the literal translation is, "Bring a child up in his way." And the question is, Is it the child's way or God's way? If it's the child's way, "when he is old he will not depart from it" might be a threat.

"Bring a child up in his own way, and when he is old he will be stuck in his own way forever." A lot of scholars think that's what that verse means. Or it might mean, "Bring a child up in His way"—that is, God's way—"and when he is old he won't depart from that."

You could go either way. Let's just assume it means what it has traditionally been taken to mean.

The nature of the Book of Proverbs is not such that they are meant to give promises without exceptions. And you can see that repeatedly, because there are places where they actually would contradict each other right next to each other.

"Answer a fool according to his folly, that he may not continue in his way" or something; and "Do not answer a fool according to his folly." That's 26:4-5. And there they are, back to back. The proverbs are meant to be general principles.

In general, if you expose a child from youth to the word of God, if you pray for him from his youth, if you involve him in a gospel-preaching church, if you surround him with love and care and model Jesus Christ for him, you have really good reason to think that God will not let his word come back empty.

But not absolutely.