Due to knowing each other for practically all their lives, they have become best friends. It is only in All Apologies that we witness the first argument between them.
As Chloe has spend a lot of time with Jasmine and Alek, due to being both Mai and the Uniter, she has had to 'blow off' Amy and Paul on several occasions, and when using her new found ability to hear things clearly from a distance, she hears Amy tell Paul that she "Might be in the market for a new best friend." When Chloe asks Amy about this, Amy is shocked that she would listen in to her friends conversations, saying that she "Obviously didn't mean it, but now, I think I just might."
Later that episode, after investigating a break in at her workplace, Chloe talks to Paul and Amy, filling them in on what had happened, but still unsure why. Amy then asks why she does't go and ask Alek and Jasmine, clearly still jealous. Chloe begins to apologize, but then stops, informing Amy that she was "pretty much done apologizing." She then goes on to tell her that although they were acting as though nothing had changed between them, it had. She wasn't the same old Chloe, and Amy needed to accept her for who she was now, faults, claws, and all.
The two of them remain close friends, and Amy even tries to help with Chloe's love life, hinting that she would be good with Alek.