The adsorption of Ag on Ge(001) surface has been studied with first-principle calculations for a coverage range from an isolate Ag adatom up to second Ag atom on the surface. Some stable sites of Ag adatom on Ge(001)-2×1 reconstructed surface were found. The scanning tunneling microscopy (STM) images of each stable or metastable site for the adsorption of a low-coverage of Ag adatom are calculated. Two-dimensional Ag islands observed by STM experiments consist of Ag ad-dimers which lie on the trenches between two Ge dimer rows. The ad-dimers are parallel to the under-lying Ge dimer direction.