A version of this argument is promoted by well-known conservative shill Jordan Peterson, you can find it here. It is fairly well disposed of in the comments to the video.
To summarize some of the main points:
- "Without God-given morality, people would lie, cheat, murder and rape all the time". This is surprisingly common. Well, the truth is, I lie, cheat murder and rape exactly as much as my morality lets me. Which is none. (OK, maybe I lie and cheat occasionally :) ) I have to wonder about people who only refrain from these things for fear of punishment. Do they ever stop to consider whether they themselves would carry out these acts if they did not have the fear of God?
- Animals have (as far as we know) no conception of god or gods. Yet there are innumerable examples of animals behaving in ways that we can only regard as loving or altruistic.
- "All our moral precepts come from the Bible". Some of them may be in the Bible, that does not mean that they came from the Bible. Most of them can be shown to pre-date the Bible, often by thousands of years.
- "If morality doesn't come from God, where does it come from?" This is actually quite easy. We know that cooperative groups that do not cheat each other tend to survive better than groups that do cheat each other. So that tendency is selected for through evolution.