For context, $A$ is amplitude, so it probably has units of length. $F_0$ has units of $\text{Force}/\text{Mass}$ and $\omega_0$ has units $\text{1/Time}$, $\lambda$ has units $\text{1/Time}$ and $\omega$ also has units $\text{1/Time}$.
To show proportionality, we would need to get $\Pi = c$ for constant $c$, so we can only have $1$ dimensionless quantity.
But if we have $F_0, \omega_0, \omega, \lambda, A$, 5 quantities, I dont see how thats possible.