I have gone to Serena almost daily for more than a year and have never received the golden axe. On a previous character, I had gotten the silver axe, but even that seems scarce lately
There are specific responses you have to choose and sometimes you don't get the choice to choose those choices. I don't remember exactly what you have to do, but you have to be really nice to her, then at the end say you really thought she would give you the axe.
Something like that, don't take my word for it. That's one things that few people understand. You'll probably need to keep trying different responses until you get it.