How to define custom force in quantum chemistry packages?
The short answer is not. You cannot define a constant force for an atom or set of atoms in a system.

If you still want to modify force constants, this is the way to do it:


Gaussian, you can modify through the keyword GEOM(IHarmonic=n), where n is part of: $\frac{n}{10^6}$ which indicates the respective force applied to the system. Its units are $\frac{Hartree}{Bohr^2}$. To more info, please refer to this link.
If you want to introduce a specific force to an atom or set of atoms in MM, the best way to do it is by parameterizing your own system into your desire environment. The most important part is that you have to have in mind that the right part of the principal ec. of any MM describes the parameters of the specific force field of your desire system or set of systems under specific conditions (i.e. aminoacids in water like OPLS)


$$\vec{F}=\frac{\partial V}{\partial x}$$


Moreover, when you talk about forces in QM you are talking about the forces acting on electrons. Not on atoms.
But, if you still want to know a way to know how the forces of electrons (From DFT) and, at the same time, nuclei are, I recommend you Car-Parrinello. Please, refer to this link
If you have a particularly interest on simulating systems in QM with specific forces acting on electrons, the best way to do it (As far as I know) is to develop your own density functional.


Update

I Gaussian input file was requested to figure out how the keyword and input is used: Geom(iharmonic=n). I did not find the respective file, instead, I give a try twice by crating two input files with different values of n. The used Gaussian input files were:

chk=Test.chk
#p opt b3lyp/6-31g geom(iharmonic=n)

Test

0 1

H   
H 1 0.9 
O 2 1.2 1 120.


and the used values of n were 2 (On the left) and 20 (On the right). The respective Energy and RMS profile of both simulations are:



Also, in Gamess, you can use these keywords:  Statpt, Force and its related keywords like Zmt. More info, Please see this link at page 2-121 and 2-131.