You can try with

c0[G_?VectorQ, \[Theta]0_, \[CapitalPhi]0_, Q_]

Beware that using capital letters for variables is not recommended in Mathematica.

Could you post an example of a complaint?

In[12]:= c0[G_, \[Theta]0_, \[CapitalPhi]0_, Q_] := 
 Normal@SeriesCoefficient[
   Sqrt[G[[1]]^2 + G[[2]]^2 - 
     2 Q Sin[\[Theta]] (G[[1]] Cos[\[Phi]] + 
        G[[2]] Sin[\[Phi]])], {\[Theta], \[Theta]0, 
    0}, {\[Phi], \[Phi]0, 0}]

In[13]:= c0[{a, b}, c, d, e]

Out[13]= Sqrt[a^2 + b^2 - 2 a e Cos[\[Phi]0] Sin[c] - 
 2 b e Sin[c] Sin[\[Phi]0]]

In[17]:= c02[{g1_, g2_}, \[Theta]0_, \[CapitalPhi]0_, Q_] = 
 Normal@SeriesCoefficient[
   Sqrt[g1^2 + g2^2 - 
     2 Q Sin[\[Theta]] (g1 Cos[\[Phi]] + 
        g2 Sin[\[Phi]])], {\[Theta], \[Theta]0, 0}, {\[Phi], \[Phi]0, 
    0}]

Out[17]= Sqrt[g1^2 + g2^2 - 2 g1 Q Cos[\[Phi]0] Sin[\[Theta]0] - 
 2 g2 Q Sin[\[Theta]0] Sin[\[Phi]0]]

In[18]:= c02[{a, b}, c, d, e]

Out[18]= Sqrt[a^2 + b^2 - 2 a e Cos[\[Phi]0] Sin[c] - 
 2 b e Sin[c] Sin[\[Phi]0]]

What function of G do you have in mind? The following works with immediate assignment, but it won't speed up calculations, I am afraid:

f[u_?VectorQ, v_?VectorQ] = u.v

Using vector notation you may write your function a little more elegantly

c02[g_?VectorQ, \[Theta]0_, \[CapitalPhi]0_, Q_] := 
 Normal@SeriesCoefficient[
   Sqrt[g.g - 
     2 Q Sin[\[Theta]] g.{Cos[\[Phi]], 
        Sin[\[Phi]]}], {\[Theta], \[Theta]0, 0}, {\[Phi], \[Phi]0, 0}]

but it only makes sense for a 2-dimensional g, and it only works with delayed assignment. Also, I think SeriesCoefficient does not need Normal.