My husband is like your family member...except that he knows all about my mental health conditions. His solution when he feels bad is to talk things over and over and over. If he doesn't feel better after that, he talks about it some more. His family does this...it works for him and for all of them. It's not how I was raised, and talking about it when I'm feeling depressed makes me feel even worse because it makes him feel like it's an opportunity to discuss how to 'fix' it (and I know he's just trying to help, but intentions mean nothing when I'm depressed). I also feel like I have to then comfort him because he doesn't like seeing me upset. I've had the following conversation with him several times, and he's finally getting it (after more than a decade):

I know you, and many people, feel better when they talk about their feelings if they're not feeling great. That does not help me. Talking it over with others makes me feel pressured and even more upset. I need you to understand that I'm not feeling great right now, and that it's OK. I will feel better when I feel better, and that change will have to come from within me. I appreciate your concern, but I'd rather be the one to approach you when I feel like talking. It's good to know that you will be there for me when I'm ready, but I need you to not pursue this until I'm ready.

Some version of this or what Agentfyre provided might work after one attempt for someone you don't have to live with.