Let $C$ and $D$ be (strict) 2-categories, let $F,G:C\to D$ be (strict) 2-functors, and let $\alpha:F\Rightarrow G$ be a (strict) 2-natural transformation.
Let now $X$ and $Y$ be objects of $C$, let $f,g:X\to Y$ be parallel 1-morphisms of $C$, and $\beta:f\Rightarrow g$ be a 2-morphism of $C$.
Consider the components $\alpha_X:FX\to GX$ and $\alpha_Y:FY\to GY$ of $\alpha$, these are arrows of $D$. Consider also the 2-morphisms $F\beta:Ff\Rightarrow Fg$ and $G\beta:Gf\Rightarrow Gg$, which are 2-morphisms of $D$.
Denoting the whiskering just by juxtaposition, is it true that $\alpha_Y\, (F\beta) = (G\beta) \,\alpha_X$?
(A way to interpret this property, if true, is "naturality condition on 2-cells".) Anyway, does this hold?
A reference would also be welcome.
The property is true. It can be found for example in "2-dimensional categories" by Johnson and Yau, explanation 4.2.4 in section 4.2, where they consider the lax natural case. For strict natural transformations, their 2-cells $\alpha_f$ are identities.
The condition asked in the OP corresponds indeed to naturality of the 2-cell $\alpha_f$ in $f$.