People here do use "quotient" as a verb: I searched for "quotienting" and got 12,890 results. [Edit: It's not as bad as I thought. Apparently I didn't understand how the search function works. When I put quotation marks around the search term I get only 207 results for "quotienting" and 64 for "quotiented".]
Why do they do this? Is it just that they want to be ungrammatical, perhaps for humorous effect? Or is there some actual confusion or ambiguity which could result from using a normal verb like "divide"?
Is the verb "to quotient" just a kind of academic slang which you can get away with only in a informal setting such as this web site or a classroom? Or is it accepted even by high class publications like the Annals of Mathematics or the AMS or LMS journals?
This is standard mathematical terminology in the context of quotient spaces and quotient groups. The use of "divide" or "division" is nonstandard and generally considered incorrect in these contexts.
One reason a new word is needed is if, for example, the group operation is addition. Then the quotient group of $G$ by subgroup $N$, written $G/N$, consists of elements $N+g$, for various $g\in G$. There is no dividing going on.