Necessity of convolution operations for product of two polynomials via brute force method
In the reference linked, in order to get product of these two polynomials, $c(x) = p(x)q(x)$, via brute force, we have to compute new coefficients via convolution $\left(c_k = \sum_{i=0}^k a_i b_{k-i} \right)$ which takes $\mathcal{\Theta}(n^2)$ time.


That's not what it says. It says firstly that convolution is equivalent to polynomial multiplication (i.e. an algorithm for convolution is also an algorithm for polynomial multiplication and vice versa); and secondly that


  The direct approach is to compute all $c_k$ using the formula above


and that this approach has complexity $\Theta(n^2)$. The FFT approach calculates a convolution (and indeed, the continuous Fourier transform produces a convolution of two functions: the convolution of two coefficient sequences is a discrete analogue).