We report the fabrication of near-vertically elongated GaN nanorods on quartz substrates. To control the preferred orientation and length of individual GaN nanorods, we combined molecular beam epitaxy (MBE) with pulsed-mode metal-organic chemical vapor deposition (MOCVD). The MBE-grown buffer layer was composed of GaN nanograins exhibiting an ordered surface and preferred orientation along the surface normal direction. Position-controlled growth of the GaN nanorods was achieved by selective-area growth using MOCVD. Simultaneously, the GaN nanorods were elongated by the pulsed-mode growth. The microstructural and optical properties of both GaN nanorods and InGaN/GaN core-shell nanorods were then investigated. The nanorods were highly crystalline and the core-shell structures exhibited optical emission properties, indicating the feasibility of fabricating III-nitride nano-optoelectronic devices on amorphous substrates.