A similar thing happened on our Altima the other day- I happened to glance out the window and saw the headlights blinking for no reason (no alarm sound was going off), and when I went outside I could hear a clicking noise. The car wouldn't start and required a jump, but then wouldn't restart after letting it run for 20 minutes or so.
It's our secondary car and doesn't move for weeks or even months at a time, so I'm used to finding the battery dead. I'm going to try replacing the battery this weekend, and I'll let you know if that fixes the problem. Not sure if this will help you, but heres hoping!