I save your calculated x11,x00,x10,x01 interpolating functions in y11,y00,y10,y01 by doing this

(*Second-correlation equations*)
...
{y11, y00, y10, y01} = NDSolveValue[...];

And then I am able to plot your function by doing this

Plot[NIntegrate[y11[t, \[Tau]], {t, 0, 450}], {\[Tau], 0, 20}, PlotRange->All]

and get the value at a point t,tau by doing this

In[15]:= y11[13, 5]

Out[15]= 0.000462099 + 0. I

Before your second NDSolve you have not assigned constant numeric values to \[Eta] or \[Alpha].

Unfortunately Mathematica does not instantly put up a message explaining exactly that, which would immediately lead you to what you need to fix, and instead puts up a message about some derivative being zero at some value of \[Tau].

If you insert

 \[Eta] = 11;\[Alpha] = 13;

or something similar prior to your second NDSolve then it appears that your error messages go away.

and how to do such a partial integration here