Sounds like an interesting family life. First off, the issue of whether there is or isn’t a God is not a subjective issue. Either God exists or God doesn’t exist, but either way, it doesn’t depend on how anyone feels or what anyone believes. That said, it’s an objective matter that nobody has settled entirely conclusively. It remains a live philosophical issue, at least in some corners, among some pretty thoughtful and well informed thinkers.
The goal of persuading an atheist there is a God or persuading a believer there is none sounds pretty intrusive. It sounds a bit like trying to take charge of another’s mind and shape it to your own will. But you just can’t change another person’s mind for them. That sounds like trying to cancel a person’s autonomy or delete their free will. Being able to make up your own mind is part of what it is to have a mind. What you can do is present the reasons you find compelling. If they really are good reason and you are talking to a reasonable person, they may find those reasons compelling as well.
But bear in mind that people usually have their own reasons for believing what they believe and they probably find them compelling as well. People usually get stuck here because we are generally not as reasonable as we like to suppose. Chances are that your reasons and the other person’s reason are not equally good. But chances are also good that neither of you is well positioned to make that determination, at least in a way the other will find compelling.
Now, there are other goals you can achieve in this conversation across differences of belief and these are quite worthy goals in themselves. In exploring each other’s reasons for believing different things, you come to understand each other better. And understanding is its own form of intimacy. It provides a foundation for mutual appreciation and caring. So your family members are doing good stuff if they are engaging in dialectic for the sake of better understanding of each other. Maybe not such good stuff if their goal is to make somebody change their mind.
When someone else offers you reasons to change your mind, you should be convinced by those reasons if they are good reasons. That’s not exactly the same thing is being convinced by the person. Nobody likes to be bullied or coerced and we often resist the reasoning others offer because of this. It can feel like a personal violation or an assault when we feel like some person is trying to change our minds.
However, we are now so much in the habit of personalizing ideas and reasons that we leave ourselves little room to consider ideas and reasons on their own merits. Framing everything in terms of “this person’s opinion” or “that person’s reasons” leaves no space for open minded critical thinking. Discourse across different ways of thinking comes to feel like a mine field of personal intrusions and attacks. I’m afraid this is now our national intellectual pandemic. We have become an unreasonable people, literally. We are not amenable to reasons because we’ve largely lost the ability to consider ideas and arguments on their own merits, independent of the willfulness of the people who offer them. When mere persuasion reigns, reasonableness dies.