>The model I am using for QBO is that there is a characteristic resonant frequency associated with the wave equation, but that an exact biannual modulation is applied.

>f″(t)+(a+qsin(4πt+θ))f(t)=0

In the code it seems you are not using the above equation but the following Hill equation

$$y''(x) + (const1+ const2*cos(4 \pi x + const3) + const4*cos(0.5195 \pi x + const5))*y(x)=0$$

I am not sure (anymore) how the solutions of Hills equation look like, but I would think that if the solution is periodic then it would have periodicities as in the given fourier expansion, so this points to a periodicity
of 1/0.5195 and not 1/0.5 (which would be the exact biannuity).

I assume that "First" plots out y(x) of NDSolve in particular I don't have Mathematica to check. Moreover in your plot you have that strange $y(x+const*sin(const x +....)
is that what you call a filter?

Where do you have the QBO data from? Did you compare that with the one in the blogpost you had linked to?