Your sentence means that you want your family to live in an apartment by themselves (rather than with another family). If YOU want to live alone in an apartment (without your family), you need to say: How can I convince my family to LET ME live in an apartment alone? How can I convince my family to ALLOW ME TO live in an apartment alone? Also, it's not a statement. It's a question and needs a question mark (?).

It’s worse than being incorrect (which is usually unimportant) because its meaning isn’t clear. Our family needs to move from our farm to an apartment, but I can’t convince them of that alone. How can I get someone else to help me convince them? I want to stop living with my family and live alone in an apartment. How can I convince them that it’s a good idea?

I think not. A family in an apartment would be living together, not alone.

Hello your sentence according to meaning is right but according to grammar we can not follow both WH and yes/now rules. When we want to use WH words we have to use rest the sentence like positive sentences. Like this: How I can this is the way to write your sentence. Thank you.