The question cannot be so simply couched. There are plenty of pagan gods out there that were horrific by today’s standards. Then there were and are many people who believed and are believers in the Judeo-Christian-Muslim God, yet whose hideous actions they try to justify using God as an excuse.
In a more subtle light, there is the longstanding argument among more respectable God-believers whether people can be directly inspired by God, or can only do so through a small minority both trained and experienced in knowledge and faith.
There are those who claim that there is complete free will; but others who claim the outcomes are foreordained; and yet others who say all our actions are predestined. Still others say that God could intervene but chooses not to most of the time; that God set the universe into motion, and has let it run on its own since.
Such arguments and debates go far beyond these, and there are no ready answers beyond belief.
Yet, there it is...
Paul explains that the laws of God are written on the hearts of man, so no man has an excuse for not knowing what is good.
So in a way the answer is “Yes, a person who does not believe in God can do good.”
But as a believer, there is an understanding that the fact that a person can do good is due to the fact that God has imbued each person with an understanding of his laws.
So the answer, from a believer’s perspective, is “No, good comes from God, whether or not a person is aware of his work in them.”