That's an interesting question. I would want to know how God himself feels about it. For example, the nation of Israel back in the Old Testament days was known for turning its back on God. God was not happy with their rejection of him, and he pleaded with them to turn back to him and honor him. So today, if we have "In God we trust" on our money, we are in some way honoring him or publicly recognizing him as God. According to Wikipedia, it is the official motto of the U.S. Back in 1956 when the U.S. started printing it on our money, the general population was in agreement, so this would be a way of honoring God by the whole country. The Bible teaches that those who honor God will be honored by him in return, and we could expect national blessings by honoring him, just like he promised to give national blessings to Israel if they would turn back to him. One cannot deny that over the years the U.S. has received blessings like no other nation. Of course, just having “In God We Trust” on our documents and money does not mean that we are completely at one with God, but it should be an indication that as a nation, we desire to be. Unfortunately, we probably aren’t as much so today as we were in 1956.
At any rate, when you ask, can we remove this quote from our money or documents?, we would in effect be saying that we are no longer honoring God if we did so. The Bible teaches that when we turn our back on him, he will leave off his interest in us and his blessing too. So I would say that, from God's point of view, it would be a very foolish thing for us to do - to remove our declaration of trust in him. It would not be a wise thing to do, especially after having already made that public commitment to him and about him back in our history. The U. S. would be making a mistake to do that. Our God is a loving and kind God who honors those who honor him. This is the path the U.S. has been on for most of its history. It could prove detrimental to us to reverse course. We should continue to put our trust in him without fear of some people's opinion, and to make room for that trust to grow deeper in us as a people as time moves along.