Maybe this:

dG = G';
subst1 = \[CapitalXi]dG[\[Xi]_] -> dG[\[Xi]]/G[\[Xi]]^2;
subst2 = \[CapitalXi]dG'[\[Xi]_] -> 
   A + B \[CapitalXi]dG[\[Xi]]^2 + c \[CapitalXi]dG[\[Xi]];
subst3 = \[CapitalXi]dG''[\[Xi]_] -> 
     D[\[CapitalXi]dG'[\[Xi]] /. subst2, \[Xi]] /. subst2 /. subst1;

a = {a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, 
   a15, a16, a17, a18, a19, a20};


U[\[Xi]_] := Sum[a[[i + 2]]*(H + \[CapitalXi]dG[\[Xi]])^i, {i, p, q}]
p = -1; q = 1;
U[\[Xi]]
eq1 = 2*(\[Nu]^2 - k^2)*D[U[\[Xi]], \[Xi]] + 
      2 (\[Alpha] k^2 \[Nu]^2)*D[U[\[Xi]], {\[Xi], 2}] - 
      3 \[Nu] k^2*D[U[\[Xi]], \[Xi]]^2 /. subst1 /. subst2 /. subst3 //
   Expand

I am trying to guess your purpose:

ClearAll[\[Alpha], eq1, \[CapitalXi]dG, coefficients, G, \[Xi], A, B, 
  H, c, k, \[Nu]];

dG = G';
subst1 = \[CapitalXi]dG[\[Xi]_] -> dG[\[Xi]]/G[\[Xi]]^2;
subst2 = \[CapitalXi]dG'[\[Xi]_] -> 
   A + B \[CapitalXi]dG[\[Xi]]^2 + c \[CapitalXi]dG[\[Xi]];

a = {a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, 
   a15, a16, a17, a18, a19, a20};


U[\[Xi]_] := Sum[a[[i + 2]]*(H + \[CapitalXi]dG[\[Xi]])^i, {i, p, q}]
p = -1; q = 1;
U[\[Xi]]
eq1 = 2*(\[Nu]^2 - k^2)*D[U[\[Xi]], \[Xi]] + 
     2 (\[Alpha] k^2 \[Nu]^2)*D[U[\[Xi]], {\[Xi], 2}] - 
     3 \[Nu] k^2*D[U[\[Xi]], \[Xi]]^2 /. subst1 /. subst2 // Expand

What is missing is a replacement rule for the second derivative of \[CapitalXi]dG