No FFL transfer is required under federal law. Unless Arizona has some ridiculous requirement to transfer handguns between private individuals within its borders, and I doubt that it does, go ahead and give it to her. I have my FFL and have run this scenario by ATF to make sure they interpret the regs same as me, and there is no federal restriction on what you want to do. I have gifted several guns to my sons with no problems. Some before, and some after, I bothered to confirm the legalities with ATF...