Pretty sure you need to sync the fobs with your zed, put the fobs in the fob slot to the right of the steering wheel, then when you remove the fob the buttons will work.
This is what timpsons showed me, i gather we are talking about a 370Z?
Hi and welcome from Bristol, i personally think that you have your work cut out with finding a good zed within your budget.
Sorry for being blunt, i think 6k plus for anything descent.
Good luck with your search