If a child was baptized as a baby without the parents' knowledge, but raised in a Jewish home, could that child feel the presence of God without knowing it?
There are too many false assumptions in this question.
One does not need to be baptized to feel God's presence. God's presence is known throughout His creation. "For since the creation of the world His invisible attributes are clearly seen, being understood by the things that are made, even His eternal power and Godhead, so that they are without excuse" (Romans 1:20). The assumption is that people are born sinful and therefore cannot know or recognize God until they become a Christian. "Behold, I have found only this, that God made men upright, but they have sought out many devices" (Ecclesiastes 7:29). All people, including those of Jewish parentage start out life without sin. For more on this, see: Total Depravity or Total Inability.
The baptism of babies also arises from the assumption that people are born sinful because they inherited Adam's sin. The idea is that an unbaptized baby is lost to Hell -- a concept never taught in the Scriptures. Rather, the Bible teaches that sins are not inherited. "The soul who sins shall die. The son shall not bear the guilt of the father, nor the father bear the guilt of the son. The righteousness of the righteous shall be upon himself, and the wickedness of the wicked shall be upon himself" (Ezekiel 18:20). For more on this, see: Infant Baptism.
The concept that someone else (a child's parents) must accept on a child's behalf is a natural outgrowth of infant baptism because the Bible clearly indicates that faith is needed prior to baptism (Mark 16:16), but an infant does not have faith -- he doesn't even have coherent thoughts. Faith is a product of learning God's teachings. "So then faith comes by hearing, and hearing by the word of God" (Romans 10:17). To sidestep this problem, the idea was that parents accept on behalf of the child, but the entire premise is false. Each person must have faith (Hebrews 11:6). No one can have faith for you.
The phrase "feel the presence of God" is a way to say that a person feels like they are saved. The problem is that God never states that He uses vague feelings to let people know they are saved. See: Is it possible to feel the presence of God when one worships God? and Does God Speak Directly to Mankind Today?
A person learns about God through the Bible. It doesn't matter where he was raised or who taught him as a child. All that matters is a willingness to learn from God. "Since you have purified your souls in obeying the truth through the Spirit in sincere love of the brethren, love one another fervently with a pure heart, having been born again, not of corruptible seed but incorruptible, through the word of God which lives and abides forever, because "All flesh is as grass, And all the glory of man as the flower of the grass. The grass withers, And its flower falls away, But the word of the LORD endures forever." Now this is the word which by the gospel was preached to you" (I Peter 1:22-25).