So you mean force constants with respect to molecular vibrations, right? That's pretty standard these days, any decent DFT tool should be able to do it. In Turbomole that would be e.g. be done with the aoforce tool, see page 48 within this tutorial:
What do you need "Force Constant Matrix" for? Do you need relaxed structures and determine the vibrational modes?
So, I do use VASP to do my DFT calculations, but you'd need a license to use it. Here's the link with its manual just in case, https://www.vasp.at/wiki/index.php/The_VASP_Manual.
On the other hand, you can try "Quantum Expresso", which is an open-source computer codes for DFT.
https://www.quantum-espresso.org/
In both software you'll be able to execute what you're looking for. To save you some time, choose "Quantum Expresso" and search for tutorials on YT.
2- After getting the forces i.e. after running the DFT code, you use DFT code output files, to get force constant matrix using phonopy.
Note: 1. we don't extract forces manually from DFT code output files. Phonopy can be used to extract forces matrix (force-sets) or force constant matrix.
2. Now whether you will get forces constant matrix or force sets from Phonopy is depends on DFT calculation methodology. If you have performed DFPT (density functional perturbation theory) calculation, you will directly, get force constant matrix from phononpy. However, if you have performed finite difference methods, you will get collections of forces called force-sets from phonopy.
3. Force_sets can be converted from phonopy to force constant matrix.