Is every homeomorphism a quotient map?
Everything you said is true, but it's kind of an abuse of the definitions.
We like to think of a a quotient map as somehow gluing things together, and a homeomorphism corresponds to the trivial case where we glue together nothing. Moreover, in a homeomorphism, we usually think of $Y$ as being a space in its own right, whereas the quotient topology is more like a structure that the function imposes on the space. (Anthropomorphizing, we might say that the function "wants" to be continuous and so it "demands" that $Y$ have a particular topology).