This is an exercise in many topology books. Here is a reference with a complete
proof: Look up Example 9.15 in the book "Knots" by G. Burde and H. Zieschang.
The Jacobian of the presentation $G(T_{p,q})=\langle x,y \mid x^py^{-q}\rangle$
is computed. It is
$$
\left( \frac{t^{pq}-1}{t^q-1}, -\frac{t^{pq}-1}{t^p-1}\right).
$$
The greatest common divisor of it is the Alexander polynomial.