If we take the square $[0,1]\times [0,1]$ and collapse the border, the resulting quotient space is homeomorphic to the sphere.
The same holds if we take the square $[0,1]\times [0,1]$ with the equivalence relation that associates $(t,0)\sim ~(0,t)$ and $(1,s)\sim~(s,1)$ (poligonal topological representation of the sphere).
Any of these spaces are diffeomorphic to the sphere? If not, is it possible to obtain the sphere as any other quotient of the square in a differentiable way?
As Rob Arthan pointed out, to ask whether a quotient space $X$ is diffeomorphic to $S^2$ one has to define a differentiable structure on that space first.
An easy way to define a differentiable structure is to pick a homeomorphism $\phi$ from $X$ onto some differentiable manifold $M$ (a sphere in your case), and declare it to be a diffeomorphism: that is, smooth functions on $X$ are defined to be precisely the compositions of smooth functions on $M$ with $\phi$.
Of course, if you do that, the question of whether $X$ is diffeomorphic to $M$ becomes moot.
If you somehow define a differentiable structure differently, the search for diffeomorphism may be less trivial (it depends on your definition), but a diffeomorphism does exist. There are no exotic $2$-spheres.