I am considering quotients of categories as in MacLane's Categories for the working mathematician, as described in the next paragraph.
Let $C$ be a (small) category and $R$ an equivalence relation on (the arrows of) $C$ such that the domain and range maps are invariant on $R$-equivalence classes (i.e., if two arrows are equivalent then they have same domain and range). Assume further that $R$ is a congruence: $f_1Rg_1$ and $f_2Rg_2$ implies $f_1f_2Rg_1g_2$ whenever that makes sense. Then $C/R$ is a category (with same object space $Ob(C/R)=Ob(C)$ as $C$).
Assume further that $C$ is a topological category: $C$ and $Ob(C)$ are endowed with topologies making all structural maps continuous. Endow $C/R$ with the quotient topology.
Is $C/R$ a topological category?
It is quite clear that the domain, range, and unit maps are continuous, however the composition is not clear because $C/R\times C/R$ does not necessarily have the quotient topology of $C\times C$ (under the obvious map). See "Products of quotient topology same as quotient of product topology".
Ronnie Brown and J. P. L. Hardy have proven that the category of topological categories has quotients in the categorical sense (satisfying the universal property) in this article, but the underlying category of this quotient is not necessarily $C/R$ as defined above.