If α and β , be the coefficients of x ^ 4 and x ^ 2 , respectively...

${\left(x + \sqrt{{x}^{2} - 1}\right)}^{6} + {\left(x - \sqrt{{x}^{2} - 1}\right)}^{6} = 2\left[{C}_{0}{x}^{6} + {C}_{2}{x}^{4}\left({x}^{2} - 1\right) + {C}_{4}{x}^{2}{\left({x}^{2} - 1\right)}^{2} + {C}_{6}{\left({x}^{2} - 1\right)}^{3}\right]$

$= 2\left[{x}^{6} + 15\left({x}^{6} - {x}^{4}\right) + 15{x}^{2}\left({x}^{4} - 2{x}^{2} + 1\right) + \left(- 1 + 3{x}^{2} - 3{x}^{4} + {x}^{6}\right)\right]$

$= 2\left(32{x}^{6} - 48{x}^{4} + 18{x}^{2} - 1\right)$

Now given that $\alpha \text{ and } \beta$ are the coefficients of ${x}^{4} \text{ and } {x}^{2}$ respectively, hence

$\alpha = - 96$ and $\beta = 36$

$∴ \alpha - \beta = - 132$