Call do_action class’s function

No, you can’t.

add_action( 'hook', array( [class], "method_name" ) ); is the right way to use a method as callback. Read about how to use methods as callback in the codex.

If you want to use a static method as callback, do it in this way:

add_action( 'hook', array( "ClassName", "method_name" ) );

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