Homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Thus there is a group action of G on X which can be thought of as preserving some "geometric structure" on X, and making X into a single G-orbit.
Quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space. Intuitively speaking, the points of each equivalence class are identified or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space.
My puzzle:
Are there Homogeneous space that do not belong to the Quotient space? Examples?
Are there Quotient space that do not belong to the Homogeneous space? Examples?
Every space is a quotient space, for example the projection $X \times Y \to X$ is a quotient map for any $Y$. So, every homogeneous space is a quotient space.
Not every space is a homogeneous space, so there are quotient spaces that aren't homogeneous. As you suggest yourself, a homogeneous space has a transitive $G$ action, and so it must look locally the same everywhere. So, if you have a space that doesn't have a transitive automorphism group then it can't be homogeneous. For example, if you glue two circles at a point this is not a homogeneous space.
You can even get examples of manifolds that aren't homogeneous spaces for Lie groups, see this math overflow question for examples.