Can I find out the change in angle between two wooden sticks connected to each other with a stepper motor?
A stepper motor can certainly be used as a rotary encoder, with the caveat that both reliability and precision will be questionable at low rotation speeds and small numbers of steps.


Question indicates need for absolute angular movement detection, i.e. a "zero detection"

No convenient way to detect zero from a stepper

Question does not specify angular resolution of stepper motor, nor angular precision required. 

A stepper with more steps (not microsteps) per rotation will provide greater precision

Speed of rotation of arm not specified

Slow rotation will result in weaker step signal, hence greater likelihood of missed steps





Now, working towards a solution:


Energize the stepper motor's coils with current-regulated DC at a small fraction of the rated operating current for the stepper. For a 3 Ampere rated stepper motor, 10 mA per coil is ample, step detection should work at even less.
Read the voltage pulses (back-EMF) generated across the stepper coils, as the stepper is rotated.
The sequence of the pulses indicates direction of rotation, the pulse count is the number of angular steps traversed, and the amplitude of the pulses relates to the angular rotation speed.
Some calibration will be required if anything other than number of steps traversed is to be detected
Do not expect lab-quality rotary encoder reliability from this solution: Occasionally steps may be missed, and some spurious steps may get detected due to EMI picked up by the coils.


This YouTube video demonstrates the use of a Hard Drive motor as a rotary encoder.

This article discusses the use of stepper motors as rotary encoders in some detail.