We have systematically measured the transport properties in the layered rhodium oxide K$_{x}$RhO$_{2}$ single crystals ($0.5lesssim x lesssim 0.67$), which is isostructural to the thermoelectric oxide Na$_{x}$CoO$_{2}$. We find that below $x = 0.64$ the Seebeck coefficient is anomalously enhanced at low temperatures with increasing $x$, while it is proportional to the temperature like a conventional metal above $x=0.65$, suggesting an existence of a critical content $x^{*} simeq 0.65$. For the origin of this anomalous behavior, we discuss a filling-induced Lifshitz transition, which is characterized by a sudden topological change in the cylindrical hole Fermi surfaces at the critical content $x^*$.