I have a strong suspicion that the Weierstrass function is non-monotonic across every interval, though this suspicion is rooted in the visualization of its generation. I don't know how to prove it, nor am I certain about whether the non-monotonicity is strict or not, though I suspect it is strict.
One way to prove strict non-monotonicity across every interval would be to prove the following, which seems to me to be a simple target:
$$\forall x,y \in \Bbb R\biggr( W(x) \ne W(y) \iff \exists z \in [x,y] \biggr [W(z) < \min(W(x), W(y)) \lor W(z) > \max(W(x),W(y)) \biggr] \biggr)$$
Perhaps another way to prove it would be via the proof for the nowhere-differentiability of the Weierstrass function, because maybe nowhere-differentiability implies strict non-monotonicity across every interval?
Weierstrass's function is famously nowhere differentiable. A monotonic function is differentiable almost everywhere, as mentioned in this Wikipedia article. Thus, Weierstrass's function cannot be monotonic on any interval.
In fact, a much stronger condition is true: Any function of bounded variation must be differentiable almost everywhere. This is known as Lebesgue's differentiation theorem and is proved as theorem 4.52 in Karl Stromberg's Introduction to classical real analysis, as well as in the above wiki link to the theorem.