Can I use a method from an existing plugin as an action hook?

It sounds like you are looking for do_action() In your method that you wrote add do_action and it will trigger your new custom action. Then you can use add_action() in the same way you use the build in actions.

https://developer.wordpress.org/reference/functions/do_action/

Example of your method in the class…

public function some_method() {
    $foo = 'puppy';
    $bar="bunny";
    do_action( 'my_nifty_action', $foo, $bar );
}

Somewhere else you can then add the call to your action…

add_action( 'my_nifty_action', 'my_custom_function', 10, 2 );

function my_custom_function( $foo, $bar ) {
    // ... your code here
}