Is that correct to say that osmosis happens with water only?
No, it is not correct because it happens with other liquids (solvents) also. 

Taking Wikipedia's definition:


  Osmosis is the spontaneous net movement of solvent molecules through a
  semi-permeable membrane into a region of higher solute concentration,
  in the direction that tends to equalize the solute concentrations on
  the two sides.


Only as extra information: osmosis occurs until chemical potentials of every substance on each side of the membrane are equal. As the substance cannot move through the membrane, the liquid (solvent) do.

According to the definition, the book isn't correct, because it says with water only, but take into account that in biology the solvent is always water. So, as an internal definition of biology (or physiology) maybe is not wrong.