The pair of amphoteric hydroxides is: 

When the OH group is bound to an element, usually a metal, the hydroxides are created. The strongest and most stable bases are made of the hydroxides of alkali earth metals like lithium, potassium, and sodium. All alkaline earth metals can be converted into hydroxides, which are less stable than alkali earth metals but still soluble.

This option, $\mathrm{Be}{\left(\mathrm{OH}\right)}_2 \mathrm{and} \mathrm{Mg}{\left(\mathrm{OH}\right)}_2$, is similarly invalid since magnesium again generates basic hydroxides and is not amphoteric as a result.
$\mathrm{Al}{\left(\mathrm{OH}\right)}_3 \mathrm{and} \mathrm{LiOH}$, First off, this is also incorrect because lithium hydroxide is basic in nature as previously mentioned. But Al(OH)₃ has amphoteric nature as it can form both acidic and basic oxides.
This choice is incorrect since $\mathrm{B}{\left(\mathrm{OH}\right)}_3 \mathrm{and} \mathrm{Be}{\left(\mathrm{OH}\right)}_2$ include acidic hydroxides of the element boron that are also not very stable.
$\mathrm{Be}{\left(\mathrm{OH}\right)}_{2 }\mathrm{and} \mathrm{Zn}{\left(\mathrm{OH}\right)}_2$, In this option, beryllium and zinc create oxides that quickly react with both acids and bases.

Hence, the correct answer is option D.