Do different phones have different accelerometer quality? Is there a way to measure it? I'm creating an Android application that basically detects the vibration of a car on streets and saves the data in a file that I'll analyze later. But I'm curious to know if there's such a thing as an "accelerometer accuracy" on different phones.

If there is, is it possible to measure the accelerometer quality of a specific phone? What are the parameters I should be paying attention when I want a phone with a very good accelerometer sensitivity and accuracy?
The quality of the sensor is mainly determined by the used sensor chip.

For a lot of devices you can find the info which sensor chip is used for providing the accelerometer/gyroscope sensor data. E.g. you can find it in a tear-down article of the device. 

With that info you can review the technical data-sheet of the used chip(s) and compare them or ask the electronic tinker scene on the net for their thoughts or recommendations.