With the understanding that being "called" means being invited by God, the answer is "Yes!"
Christ used the words called and invited nearly interchangeably in the parable about those summoned to a wedding, for example (Matthew 22:1-14). Calling is the first phase in the process. Those called must decide whether and when to accept that invitation.
Calling necessarily precedes baptism. As the Holy Spirit was with the disciples before it was in them (John 14:17), so the Holy Spirit works with called individuals. However, being called does not equate to actually receiving the Holy Spirit. That takes place after baptism with the laying on of hands. Before a person is baptized, he must repent, exercise faith in the sacrifice of Christ for the remission of sins and willingly choose to follow God. Those who respond to God's calling are described as "chosen." Those saints who will rule with Christ are "called" and "chosen" and "faithful."
Can a child be called? Scriptural evidence leaves no doubt that a child can have a relationship with God. Examples include Samuel (1 Samuel 2:18, 26), Josiah (2 Chronicles 34:3) and Timothy (1 Timothy 3:14-15). Young people can understand spiritual concepts (Proverbs 3:1-3, Psalm 34:11).
Parents are told to teach their children God's way (Deuteronomy 4:9-10) and to follow Abraham, "the father of us all" (Romans 4:16), who commanded his children to keep the way of the Lord (Genesis 18:19).
On Pentecost, after Peter's call to repentance and baptism, he said, "For the promise is to you and to your children, and to all who are afar off, as many as the Lord our God will call" (Acts 2:39). God's promise to the offspring of the converted is a special blessing. They are in a unique category, holy or set apart by God (1 Corinthians 7:14). Though they are not converted, they do have a special access to the knowledge of the way of God.
The way children growing up in the Church are called will differ from what most adult converts experience. A child's calling is gradual and becomes more personal as he or she grows and matures. Experience shows that, as with all people called, some young people who grow up in the Church relate to and respond favorably to God's way of life, and some do not. Many factors determine the type and degree of positive exposure to God's way. These include the child's age when the parents are called; influence of friends, peers and popular culture; personality, aptitudes, interests and temperament; example of parents and other family members; and influence of the congregation.
The greatest positive factor in children's lives should be the parents. Paul told parents not to "provoke your children to wrath, but bring them up in the training and admonition of the Lord" (Ephesians 6:4).
The Church also bears certain responsibilities for spiritual child development. Jesus commissioned the Church to make disciples (Matthew 28:19-20). What more fertile ground exists for making disciples than that of our youth who have already been called by God? The entire Church must be involved in this work.
But what about children who choose not to respond to God's calling? They have not necessarily forfeited their only opportunity to receive eternal life. Only God knows the heart or mind of each individual. Those He deems to have not received an appropriate opportunity for salvation now will assuredly receive one later.
God's calling is not a one-time event, but a process. It involves ongoing interaction between a loving Father and His children. Some choose to follow the path of the prodigal son, a tragic choice, but God is patient, "not willing that any should perish but that all should come to repentance" (2 Peter 3:9).
More details will be available in a video seminar being prepared on this subject.