Effectively, yes, but actually no. At least I wouldn't know how you'd do that.
A quaternion stores cos(½?) as one of its components, and the cosine function has a period of 2?.
Which means you can unambiguously represent a rotation of +/- ½·2? = +/- ? or +/- 180°.
But of course rotating by -1° is the same orientation as rotating by +359° so effectively you can get any orientation.
Ok. So given an XY coordinate that lies on a circle, is it possible to determine the angle? Applying your logic, the answer would be no, because:
angle = acos( X )
And that would only give you +/-90 degrees.... whereas the correct answer is yes, so long as you use both components, i.e. atan2(Y, X).
So sure, if you're going to completely ignore the existence of the sin(angle / 2) in addition to cos(angle / 2), then you will indeed be able to ignore the fact that a quat can represent +/-360 degrees. But you know, as I said before, if you use a crap SLERP implementation that performs a quat-negate when the dot product is less than zero, then you will not be aware of the other half of the hemisphere. If you were to change your slerp from:
if( dot(Q1, Q2) < 0 )
to
if( dot(Q1, Q2) > 0 )
your quats will always take the longest path, i.e. the one greater than 180 degrees. FYI, if you have a quat Q, that represents a rotation of 1 degree, it's negate represents the rotation of -359.