No. The ArcGIS License Manager should be of same or later version than ArcMap (ArcGIS Desktop) version.
You will need to upgrade ArcGIS License Manager to 10.5 or later to run ArcGIS Desktop 10.5 on a client.
There are multiple critical security issue in older versions of the FlexLM, which is used in Esri's License Manager.
I'd install the 10.6 version of the LM, which should be backwards compatible to all earlier versions of ArcGIS Desktop.
Hi Michael Volz -
They don't *NEED* to per-se, at least not in the security sense. In this case, I recommended moving to 10.6 to support any potential clients that might be upgraded past 10.5.x, since it's backward compatible.