Does belief in God lead to keeping the commandments?

I will start with a comment. In my opinion, your question is similar to someone who says, "I know it is immoral to murder, but why not murder?" If you ask such a question, it means that you do not really know that it is immoral to murder. Anyone who knows understands that if it is immoral then it is not right to do so. If you ask such a question, then at most you know that people think that there is such a thing as "morality" according to which it is forbidden to murder, but you yourself do not know or understand it at all. The point is that "morality" by its very nature is something that must be observed in practice. Knowing that morality prohibits something is not neutral knowledge, like knowing that the table in front of me is black. Moral knowledge includes within it an obligation to perform (in analytic philosophy, this is called a prescriptive rather than a descriptive claim).
In my opinion, the same is true for those who say that there is a God who gave commandments and ask why they should be kept. The claim "God commanded X" is prescriptive and not descriptive, meaning that it includes an obligation to perform. Those who ask such a question do not actually know that God commanded, but they know that there are people who think that God commanded. Or they know that there is someone who commanded but they do not know that it is God (and only think that others see him as God). God, by his very nature, has authority and his commands are valid. Therefore, in the Torah, the word "God" also describes judges. A judge has formal authority; his decision must be obeyed by virtue of being a judge, whether he is right or not. This is exactly the case with God and his commandments.
For more details, see my article here on the site about philosophical gratitude.