Bigg EZ

So I bought my 03 350z about a year ago, and it never gave me any problems until about 2 months ago. The problem is very random and inconvenient. it happens when I drive my car for a little while and park it so I can get gas, go into a store, etc. when I turn the key the engine turns over once and thats it. Then if I wait few hours it starts right up. All the lights on the dash come on and I know its not the battery itself because I bought a brand new one when this first happened, and it didn't make a difference. I also just got a new starter put in so I don't think thats the problem either. The engine isn't overheating so I have no clue what the problem is. I've taken in to the shop multiple times but by the time they had it towed up there, it started working.
If you've had this problem or know anything about it please let me know