The Bourns EN series spec sheet says that there are various models available from 25 to 256 cycles per revolution. So even with the highest-resolution model, it isn't quite 1 degree resolution.
The resolution can be improved just by how you detect and process/count the quadrature outputs.
A 256 cycles (often called steps) per rev can be doubled to 512 steps/rev by interrupting on change (both edge transitions) of channel A or doubled again to 1024 steps/rev by interrupting on change of both A and B channels. So resolution of the encoder in not a problem. The mechanical attachment to the hinge pivot maybe a challenge however.
Lefty