First, I saw in the billing section how much I pay for NAT Gateway, I need to understand exactly what I'm paying for, I suspect that the git checkout (we use GitHub) from our instances takes the most of the cost, but I need some way to prove / see exactly the traffic I pay for, is it possible? If so, how?
NAT gateway does not publish any information about how much data is processed by source/destination. You can deduce it by searching VPC flow logs. This documentation may be useful.