Do different phones have different accelerometer quality? Is there a way to measure it?
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.