Alice and Alex are very respectful of each other and seem to show some concern for one another, but they are not friends (according to Alex). They are mutual acquaintance which means that if they have a common enemy they may work together but in most part, they try not to get into each other hair.
In story didn't mention anything about how close they are but it certain that Princess Alice interesting about Mariya Yuri. It can be seen by Alice teach Yuri how to develop and use her power, it can happen because Alice and Yuri have the same disposition (power).
By the end of the story (Final War Arc), after Black Prince Alex banish into the parallel world, Alice support Godou as best ask she can. The support Alice gave is like making secret medicine him, introduce him to Prometheus and lead him to the Bearer of Fate territory.