First we note that
$ S = \sum_{k \ge 0} \frac{1}{k^2+k-\alpha} = \sum_{k \ge 0} \frac{1}{(k+1/2)^2 - (4\alpha+1)/4} = 4\sum_{k \ge 0} \frac{1}{(2k+1)^2 -a^2},$
where $a=\sqrt{4\alpha+1},$ and so
$ S = 4 \sum_{m \textrm{ odd}} \frac{1}{m^2 - a^2}.$
Now we use the well-known cotangent identity (at the bottom of the page)
$\pi\cot(a\pi) = \frac{1}{a} + \sum_{m=1}^\infty \frac{2a}{a^2-m^2} . \quad (1)$
Replacing $a$ by $a/2$ and dividing by $2$ gives
$ \frac12 \pi\cot \left( \frac{a\pi}{2}\right) = \frac{1}{a} + \sum_{m=1}^\infty \frac{2a}{a^2-(2m)^2} . \quad (2)$
Subtracting $(2)$ from $(1)$ gives
$ \pi \cot(a\pi) - \frac12 \pi\cot \left( \frac{a\pi}{2}\right) = \sum_{m=1, \textrm{ odd } m}^\infty \frac{2a}{a^2-m^2} $
but $\cot(\theta) -\frac12 \cot(\theta/2)= -\frac12 \tan(\theta/2)$ and so
$ \frac{\pi}{4a} \tan \left( \frac{a \pi}{2} \right) = \sum_{m \textrm{ odd}} \frac{1}{m^2 - a^2},$
from which the result follows setting $a = \sqrt{4\alpha + 1}.$
EDIT: The cotangent identity is proven here.
EDIT2: An easy way to discover the cotangent identity is to take logarithms of the following product formula for $\sin \pi x$ and differentiate wrt $x.$ $\sin \pi x = \pi x \prod_{n=1}^{\infty} \left( 1 - \frac{x^2}{n^2} \right)$