1. Use the definition of $Q$

Given

$$Q = P A P^T$$

and we need

$$P^T Q^{2007} P.$$

Since $P$ is an orthogonal matrix (it is a rotation matrix), we have

$$P^T P = I.$$

So,

$$Q^{2007} = (PAP^T)^{2007} = P A^{2007} P^T.$$

Therefore,

$$P^T Q^{2007} P = P^T (P A^{2007} P^T) P = A^{2007}.$$

So we only need to compute $A^{2007}$.

---

2. Compute $A^n$ for

$$A = \begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$$

Write

$$A = I + N,$$

where

$$N = \begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}.$$

Now,

$$N^2 = 0.$$

Hence,

$$A^n = (I+N)^n = I + nN$$

by the binomial theorem.

So,

$$A^{2007} = I + 2007N = \begin{bmatrix}1 & 2007\\0 & 1\end{bmatrix}.$$

Thus,

$$P^T Q^{2007} P = \begin{bmatrix}a & b\\c & d\end{bmatrix} = \begin{bmatrix}1 & 2007\\0 & 1\end{bmatrix}.$$

Therefore,

$$a=1,\quad b=2007,\quad c=0,\quad d=1.$$

---

3. Evaluate the required expression

$$2a+b-3c-4d = 2(1) + 2007 - 3(0) - 4(1).$$ $$= 2 + 2007 - 4 = 2005.$$

---

4. Check with options

The value is

$$\boxed{2005}.$$

So the correct option is D.