My mother was adopted about 64 years ago and she isn't interested in finding any of her relatives,however I would like to know them how can I find out this information for free?

It is very possible that you will not find out, even if it was 20 years ago, it is next to impossible to find, you post on dozens of sites and hope the other side ( parent or child) is trying to find you also.

Plus without her help to give you information, I see no way.

Most likely her bio parents are either very very elderly or passed away. Can not see them on the internet looking for their lost child.

So even if you were going to pay for the info, I doubt they would find them.