The term Brownian Motion is defined by Wikipedia as "random motion of particles suspended in a" liquid or gas. Thus it is not defined for the vacuum.
It is explained as interaction between the particle and randomly moving atoms or molecules around it.
I would like to know if it has ever been tested if a similar phenomenon might occur between a particle and the vacuum.
What I am asking for more specifically is if it has been tested if some quantum entity / event of the vacuum might interact with a particle in a similar way. (For example spin networks of Loop Quantum Gravity or virtual particles)
This is a really interesting question. I think it is first worth noting that Brownian motion is not unique to particles suspended in a gas or liquid, but rather random movement in any medium. It is the interaction between these particles that eventually causes movement that we see as random. These systems are chaotic, which is why the phenomenon of Brownian motion arises. However, virtual particles appear in pairs of normal particles and anti-matter particles and are attracted to each other. So, when they collide, it results in the anhelation of these particles. Because of this, I do not think Brownian motion will occur. However, I am unsure of it as far as quantum loop gravity, as I am not very knowledgeable in that area.
I found the answer to my question stated clearly on Wikipedia:
In July 2020, scientists reported that quantum vacuum fluctuations can influence the motion of macroscopic, human-scale objects by measuring correlations below the standard quantum limit between the position/momentum uncertainty of the mirrors of LIGO and the photon number/phase uncertainty of light that they reflect.
See Quantum Fluctuations which cites several sources including "Nature". The answer to my question above is "yes".