I have an 04 Quest that had similar problems. Please know, I am no mechanic, but I will be glad to share my simple knowledge. First, my battery was not secured, and my husband had to slam on the brakes one day at lunch. The car seemed fine; he drove back to work, and backed into the parking space. Went to leave 3 hours later, the van would Not Start. We had a few problems with the neg. battery terminal staying tight so that was the first thing we looked at. Pushing, pulling and tightening it, did not help us this time. We figured the battery shifted, when he slammed the brakes on, and that must have caused a short somewhere. Bottom line, I got it started (several days later) by testing all passenger compartment fuses. (That is what it’s called, but it is actually on the driver’s inside dash, about knee level, above emerg. brake).
There were several fuses that were blown, but the problem was the “shift lock” fuse. If I am not mistaken, the shift lock is the feature that will not allow you to start your vehicle (with automatic transmission) without the vehicle being in PARK (or neutral). I tested the fuses with a light meter (car turned over but not started). Even after replacing this one (“shift lock”), it still said it was bad. I read somewhere that I could try pushing & pulling the fuse into its position several times. This sort of “cleans” the fuse holder/slot from any debris, or buildup. Viola! THE CAR STARTED!
Just after this, my car would stall out, with NO warning. I was driving on the freeway the first time. After I sat on the side for several minutes, I tried to restart, and it got me home. That time. My check engine light had come on so I went to Auto zone to have them check for any codes that came up. It was the Cam Shaft Position Sensor. Once that was replaced, it has run fine. I do not know if both problems were related to slamming on the brakes, so I wanted to mention both.
I hope helps (in some small way) to resolve your problem. Good luck.