The ship don't need to spin relative to anything, but as you note it could be tricky to determine exactly how quickly it's spinning without visual references.
The method your protagonist would use depends on which tools are available.
If the protagonist can move about the entire ship, measure all distances and measure acceleration everywhere, there are other approaches, but I don't see why you necessarily could. For all I know, the protagonist sits in a capsule joined to a counterweight by a wire, the two spinning around their centre of gravity. In that case, there's no way of knowing any ofit's harder to know the relevant measurements with any accuracy, especially if there are no windows in that direction.
TheIf you can measure distances, forces and angles (where the latter can be accomplished by distance measurements and trigonometry), you could still find various "vertical" lines and use trigonometry to see how far away they intersect. You need to be able to determine or a kilogram or a Newton to start out with, to have some reference.
If you can't measure the centrifugal force accurately, but can measure time, the easiest way of determining that you are in fact spinning is to measure the Coriolis force. Lift an object, drop it and note how far off from the point "directly below it" it lands. The point "directly below" could be determined by tying a weight to a string, and you could simply note how fast the falling object deviates from the plumb line.
The math isn't terribly difficult, but the linked Wikipedia page is a nice place to start and get some formulas!
If the spinning is too slow, in terms of revolutions per second, your protagonist could put up a Foucault pendulum. This would give her a rather accurate idea about how fast she's spinning.