A High Altitude Platform System (HAPS) is an emerging technology that can potentially bring connectivity to areas that are not partially or totally covered by cellular networks. However, allocating certain frequency bands for the HAPS alongside wireless Fixed Service (FS) imposes some restrictions on operating the HAPS systems to ensure no interference occurs between the two systems (HAPS and FS). This paper presents an analytical study of the spectrum sharing between the HAPS and the FS in millimeter waves, namely in 38- and 47-GHz bands. Some potential and significant interference scenarios have been applied in order to investigate the spectrum-sharing situations in urban and suburban areas. The Carrier to Interference plus Noise Ratio (CINR) has been adopted as the main criterion to assess the performance of the HAPS. It is found that the HAPS and FS systems can simultaneously share the 38- and 47-GHz bands with some restrictions to HAPS altitude, allowable CINR, and location of the HAPS user. These restrictions differ depending on the area coverage type.