Personally, the green looks a little off for alexandrite, but possible. I’d try to eliminate color-change garnet first, however, just in case.
Natural alexandrite and natural color-change garnet are both valuable stones given that they’re good quality in terms of color and the cut is good (looking at just these pics it’s hard to est cut/color/clarity qualities), but I believe the alexandrite would be the more valuable of the two if you could make a distinct identification or have it verified by an expert. Pricing is less my area than identification, but natural alexandrite can be more expensive than equivalent-size diamonds depending on quality of the stone.
You could use RI to eliminate alexandrite, but not garnet (as their RI overlap and color-change garnet has a broader range of possible RI). It’s possible you may be able to eliminate garnet by determining if it has double refraction (determining birefringence) as garnet should not be doubly refractive.
Only other thing I can think of is maybe diaspore (csarite)? Not sure how likely that is, but still, can’t be ruled out just from a picture since the color-change matches what’s been found in Turkey.
Sorry for the rant! Hope that helps a little!