Is this set convex ?2 Is this set convex for every arbitrary $\alpha\in \mathbb R$?
$$\Big\{(x_1,x_2)\in \mathbb R^2_{++} \,\Big|\, x_1x_2\geq \alpha\Big\}$$ 
Where $\mathbb R^2_{++}=[0,+\infty)\times [0,+\infty)$.

Yes, it is. You should separate the case $a<0$ (when it is the quadrant) and $a>0$ when it is bounded by an arc of a hyperbola.
