I don't think there is much for me to elaborate beyond the title question: "What is the terminology for a product of a ring with a group (like the quaternions) or (more generally) with a monoid (like a polynomial ring)?"
I know the quaternions require a field, and fields are rings, so the example holds. Also, it is popular example and was the first to come to mind. Though, if you think I should put something else instead or in addition to, please let me know.
So, if you know of terminology for any of the four products of a field or (preferably) ring with a group or (preferably) monoid, please let know each one you know. (In case of confusion, the four products I'm referring to are field with group, field with monoid, ring with a group, and ring with monoid.)
If you can think of anything useful for me to elaborate, please let me know.
Edit: Right before I was about to submit this, I remembered some of the answer. A product of a ring with a group is called a group ring, right? Ring with a monoid is a monoid ring, right? Is there terminology for the other two?
I do not think anyone calls the group ring $R[G]$ “the product of a group and a ring”, but evidentially that is what you are talking about. Yes, the same construction using a monoid instead of a group Is called a “monoid ring $R[M]$.”
When the ring is a field, you could also call it “the group algebra $k[G]$” or “the monoid algebra $k[M]$, respectively.
There are also semigroup rings and semigroup algebras which do not take much imagination to define starting from this point.
One more comment to address this: the quaternions themselves are not a group ring over a nontrivial group. The problem is that it is a simple ring, and $R[G]$ is not simple unless $|G|=1$ and $R$ is simple.
I imagine you thought $\mathbb R[Q]$ is related to $\mathbb H$, where $Q$ is the quaternion group. It is, but not directly: $\mathbb H$ is a quotient of $\mathbb R[Q]$ and not a group ring itself.