For the piezoelectric tensor calculations, you can take a look at this tutorial,

https://docs.quantumwise.com/tutorials/piezoelectric/piezoelectric.html. There is no analysis object in QuantumATK that computes the flexoelectric tensor.

So, you may consider looking through the literature to see how that tensor is "manually"" computed using first-principle methodology. For example, see "Direct approach for flexoelectricity from first-principles calculations: cases for SrTiO3 and BaTiO3, Tao Xu1, Jie Wang1,2, Takahiro Shimada2 and Takayuki Kitamura2, Published 24 September 2013 • 2013 IOP Publishing Ltd, Journal of Physics: Condensed Matter, Volume 25, Number 41. I think you have all the capability of doing it in QuantumATK. Some scripting will perhaps be required.