How to connect an Output to a Pullup Input of another arduino
Connecting any signal that is within the voltage range of the receiving Arduino is safe. So if you use Arduino UNOs or similar with a 5V operating level, you can connect any signal between 0 and 5V to an input. It does not matter whether the input is in pull-up-mode or not. So connecting an output to an input of another Arduino (or even of the same Arduino) is safe. In addition to the signal values, you do need to connect the ground of both Arduinos.
The only thing you must not do is connect two outputs together (because if one of them is high and the other is low, you have a shortcut that will potentially fry one or both of the boards). And you would need to be careful if you used different types of Arduinos, like a Uno and a Due, because they run at different voltage levels.