Sapphire resonators with $YBa_2Cu_3O_{7-delta}$ (YBCO) endplates have provided a way to realize extremely high quality factor due to the extremely low dielectric loss of sapphire and conductive loss of YBCO films, which enables to measure the low surface resistance of superconductor films at microwave frequencies. We present microwave properties of HTS sapphire resonators designed for measuring the surface resistance of HTS films at millimeter-wave frequencies by using the two-resonance mode dielectric resonator method. Despite enhanced surface resistance ($R_S$) of YBCO films due to the quadratic frequency dependence of the $R_S$, the unloaded quality factor ($Q_0$) of the $TE_{021}$ mode sapphire resonator still appears to be well above $1;{ imes};10^6$ at a mm-wave frequency of 38 GHz at 10 K. However, it appears that the $TE_{012}$ mode $Q_0$ is unexpectedly low despite that the corresponding resonance peak looks uncoupled with parasitic modes. We discuss possible reasons for the unexpected results using the surface resistance at the $TE_{021}$, $TE_{012}$, and $TE_{011}$ mode frequencies.