We are planning to upgrade ArcGIS Desktop 10.2.2 and ArcGIS Server 10.2.2 to a newer version. There are only GP services and Geodata services in our environment and there is no Portal for ArcGIS. I would like to know 10.5.1 or 10.6.1 (for both desktop and server) will be ideal in this case.
Good job - it's very important for ALL users of ArcGIS Enterprise software versions 10.3.x and below to be considering an upgrade strategy, as ArcGIS 10.2.x is already in mature support, and 10.3.x will be going into mature support in December 2018. When a product reaches mature support, patches and service packs are no longer built for that version.
10.6.1 is an excellent choice for your upgrade path, and I wholeheartedly endorse this move.