Is there any such thing as a force quantum? Consider two massive particles at a certain distance $R$ from one another in $\mathbb R^3$.
The two particles attract each other through a certain interaction that decreases with $R$ : $F_{1\rightarrow 2}\propto 1/R^\alpha $.
Is there a distance beyond which the particles will not start getting closer to each other because the force felt by each in this case would be below a certain treshold (a force quantum) which makes for effective 0 (zero) interaction ?
 A: No. There isn't. Basically if you do the following:
$$
\lim_{r \to 0} \; F= \lim_{r \to 0}\; \frac{\mu}{r^\alpha} = \infty \; \; \; \; \; \; \; \; \forall \alpha>1
$$
This means that the force felt by the particle blows up to infinity and they will "collide".
If you want them to be nearby but not collide you will have to make at least one of the particles to rotate; In other words, angular momentum is needed. For now on I will talk aboud the case in which $\alpha = 2$, for simplicity, but it could be easily extended to other values (See: Effective Potential).

Classical Case:
When you add up the angular momentum, the energy of the system looks like this:
$$
E = \frac{1}{2}m\dot{r}^2 + \frac{L^2}{2mr^2} - \frac{\mu}{r}
$$
You could find an stable orbit at:
$$
r = \frac{L^2}{\mu m}
$$
If for example, this particle is an electron and is orbiting an atom, this rotation will make this electron lose energy because of electromagnetic radiation.

Quantum-Mechanical Case:
Electromagnetism:
Borh realized that stable orbits without radiation could exist. Then this was later refined by Schrödinger who came up with his famous equation:
$$
i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2 \psi \; - \frac{e^2}{4 \pi \varepsilon_0 r}
$$
This is the Schrödinger equation for hidrogenoid atoms. Solving this equation gives the energy levels of the orbitals of hydrogen. There is a minimun level of energy; the state in which the electron cannot go any lower in the atom:
$$
E_1 = \frac{m_e e^4}{8h^2\varepsilon_0^2} \approx 13.605 eV
$$
If you want to picture this ($r$ in the $x$-axes and $V(r)$ in the $y$-axes):

The electron is in this well, "oscillating" around the nucleus. The angular momentum of the particle guarantees that it won't crush into the nucleus (that's why the potential  goes to $+\infty$ at $r = 0$) and the quantization of the angular momentum (each color respresent different values of $L$) only allows the electron to be at specific distances from the nucleus.
Strong Nuclear Force:
There is also other case: The strong nuclear force. If protons and neutrons are nearby they won't repel and stay "glued" (See: Gluon) because of this force. It's a much more complex force compared to the electromagnetic force. You could, again, picture it out with the following:

Imagine this graph as if you dropped a marble that rolled down this curve. If you see, this force is strong when you are near to the particle but no so when you are very far away (in the limit, it goes to zero).
There is never a zero interaction, the things just stabilize and are mantained that way because there is equilibrium. Protons and Electrons are exchanging photons to mediate the electromagnetic force and Protons and Neutrons exchange gluons to mediate the Strong Nuclear Force without stopping at any time.
I don't know if I completely understand your question, but I hope this helps.
A: The model in which we usually speak of forces is classical mechanics. And classical mechanics treats the $\frac{1}{r^2}$ rule as exact.
There is also quantum field theory, which is a more accurate description of reality. But QFT doesn't atrribute definite positions to particles, so this question would be meaningless there.
We could instead talk about a partly quantum - partly classical model in which particles have a definite position, but the force-field between them is Quantised. Is there a threshold distance for the electromagnetic force in such a model? There isn't.
In such a model, the electromagnetic field will become an operator field. The value of the electromagnetic field will be probabilistic. But, if we take the expected value of the quantum equations, they will look identical to the classical equations because of Ehrenfest's theorem.
So, the particles will get accelerated by the expected value of the electromagnetic field. Since the expected value obeys the classical equation, there is no distance threshold to it.
Remark The force between the two bodies isn't applied instantaneously at a distance. The field takes time to propagate. So it should make sense that there isn't a distance threshold for the two particles to be able to affect each other.
A: The Pauli exclusion principle can play the role of such an effective quantum force that prevents particles from interacting. This is a well-known effect for quantum gases of polarized fermionic atoms, which makes them hard to cool.
This happens at least for short-range interactions, which include power-law potentials that fall off fast enough at large distances -- the van der Waals interaction is among them.
Indeed, at low enough energy, scattering happens in the s-wave channel, which is symmetric under the exchange of particles. If the fermions are polarized (all the same spin), their orbital wavefunction must be odd exchange, and the fermions cannot be in s-wave. Hence they do not scatter anymore and are effectively non-interacting.
A: If you consider the colour force, it gets weaker initially as two quarks approach each other and then becomes repulsive at very short distances and does not become zero.
