If a christian, or someone with belief in a god asked me for forgiveness I would say to them that they and the universe are the one and the same thing, and that god and the universe is one and the same thing and that they and the god are too the same thing by that rational. Then i would ask them to pray to their god and ask for forgiveness. Hopefully they would then realise that when we ask forgiveness of someone, we are not asking to make them feel better, but to make the person themselves feel better.
If I feel bad about something i have said or done to someone i should first forgive myself and change and endeavour not to do that again which made me sorry in the first place, for what is the purpose of forgiving if the offender is going to re offend. Once i have forgiven myself then I should approach the offended and explain that I am sorry.
This is the way of my utilitarianism, the good of the one outweighs the good of the few or the many.