Formula for non - active power of capacitors connected in a star?
I don't know the proof( we were just given the formula right away) and
I am not able to identify if I am using it correctly.

\$Q_c=ω∗C∗V^2\$ is the reactive power (Q) of a capacitor (C) connected across an AC voltage source (V). It's related to real power in a resistor (\$P = \frac{V^2}{R}\$) but, because we know the reactance of a capacitor is \$\frac{1}{ωC}\$, the reactive power formula becomes as you show it.

does Qc represent the non - active power that the whole star produces?

No, it's the single phase reactive power of a capacitor across a voltage source. If you have three capacitors in star formation and you only have the line voltage then, the reactive power for each of those capacitors is found using \$\frac{V}{\sqrt3}\$.
In fact you might as well convert your first formula to a single phase apparent power by dividing it by 3 then solving as a single phase problem.