What is the age of accountability or the time when someone becomes responsible for the gospel?
Salvation is of God
When Jesus introduced the gospel, He came, “saying, “The time is fulfilled, and the kingdom of God is at hand; repent and believe in the gospel” (Mark 1:15), so unless a person is brought to repentance by God (Acts 5:31, 11:18; 2 Tim 2:24-26), and been regenerated by the Spirit of God to believe (Eph 2:1-2), they don’t know what to repent of or what to believe in. Sin to the sinner is nothing to worry about. They don’t give it a second thought, but for the one being called by God, the Spirit convicts them of their sins (John 16:8) and makes them see their need for the Savior, so salvation is fully an act of God (Psalm 3:8; 62:1). This means we cannot talk our children (or anyone else) into believing. We can simply share the gospel and let the Spirit of God work. It takes the Spirit of God and the Word of God. Jesus wants us to have child-like faith, meaning that we believe and have faith that what we are told to be true in the Bible is true indeed; particularly, what’s written about Jesus Christ. Sometimes, it’s easier for children to believe than the more cynical adults around them, and this explains why so many were saved as a child and why so many of them were saved at a very early age, but clearly, it’s God’s choosing and it is His timing (John 6:44), so what is the age of accountability? 
Age of Accountability
The Bible doesn’t give us a specific age where someone will be held accountable for knowing the truth about the gospel and yet choosing to reject it. There is an age of accountability when a person becomes responsible for knowing God and knowing right from wrong, but we don’t know what age that is. A child is not of age to drive a car because they’re not responsible for that knowledge at their age, so when do we know a person will be held responsible for not trusting in Christ? What about those with special needs and are adults? And at what point does that knowledge come to a person, and therefore they’ll be held responsible for it? I have no idea because each person is different. One person matures more quickly than another, but intellectual knowledge won’t bring us to the knowledge of Christ. Spiritual things like that are foolishness to us until the Spirit of God makes sense of it, but at what age are they responsible for rejecting Jesus Christ? There may be many young adults who may not be responsible even now, but we just cannot give an age. It appears that the age of accountability is not the same for everyone, and it’s not a hard-fast rule like “At age 10” or “At age 18.” We just cannot say for sure.
Many are called by God late in life. Many others are saved in prison or when faced with life-threatening illness or disease, but the vast majority of Christians are saved as a child. Their parents had no idea when they would be held responsible. That’s something that I believe only God knows, so the most important thing is, God knows. Just before the nation of Israel was to cross over into the Promised Land, they refused to go because of the bad report they heard from 10 of the spies, but Moses knew that the children could not be held responsible for their parent’s disbelief. He said, “[A]s for your little ones, who you said would become a prey, and your children, who today have no knowledge of good or evil, they shall go in there. And to them I will give it, and they shall possess it” (Deut 1:39). In other words, they were not held responsible for not being able to cross over into the Promise Land. It was their parent’s disbelief that kept them from entering the Land, but the children must have believed the Lord because they did cross over after 40 years. God reminded their parents that “your little ones, who you said would become a prey, I will bring in, and they shall know the land that you have rejected” (Num 14:31). Deuteronomy 1:39 and Numbers 14:31 also tell us that children who are not at the age of knowing “good or evil” can’t be held responsible for their disbelief.
When Babies Die
When David and Bathsheba lost their baby, they grieved as any parent would, but David held out hope of seeing the child again. After the child died, the king’s attendants asked David why he wasn’t fasting anymore now that the child was dead, so David told them that “now he is dead. Why should I fast? Can I bring him back again? I shall go to him, but he will not return to me” (2nd Sam 12:23). Job also knew that there was more beyond the grave. He said, “I know that my Redeemer lives, and at the last he will stand upon the earth” (Job 19:25), so “If a man dies, shall he live again? All the days of my service I would wait, till my renewal should come” (Job 14:14). Clearly, he shall live again, and so too, I believe, will babies, infants, still-born’s, and even those with severe mental disabilities or disease. Perhaps they never had a chance to know the gospel or had the mental capacity to know what it means to believe. God alone knows, but those who die in infancy or before, or at a very young age, are not lost. Their parents will see them again, but only if they too believe in the Lord.
Since the Bible is silent on the age of accountability, so will I be. I cannot imagine what that age is. Besides, there’s no way to now when a child reaches the age of accountability before God. God alone knows the heart. All we caregivers can do is to love them, pray for them, and nurture them (in the Word). Leave the rest up to God. He knows the age of their accountability. We cannot know this and have no business trying to guess what age it is. What is not clearly revealed in Scripture cannot be clearly revealed to us.
Here is some related reading for you: Where do Babies go if They Die? Do They go to Heaven? 
Resource – Scripture quotations are from The Holy Bible, English Standard Version® (ESV®), Crossway Bibles. (2007). ESV: Study Bible: English standard version. Wheaton, Ill: Crossway Bibles. Used by permission. All rights reserved.