Alice and Alec are very respectful of each other and seem to show some concern for one another, but they are not friends (according to Alec). 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).
At the beginning, Alice was surprised by Godou bold nature and desire for battle (under the spell of Lancelot) but after the reveal, Alice started to see Godou in a better light and tried to aid him and his friends with whatever she can. By the end of the story, after Black Prince Alex was banished into the parallel world, Alice started to support Godou to help end the battle and return Alec back. The support Alice gave is like making secret medicine him, introduce him to Prometheus and lead him to the Bearer of Fate territory.