Is it possible to convert a settler ship into a settlement without moving and then build or is it considered a move?
Converting a settler to a settlement is done as part of the movement phase, and can't be done during the build and trade phase.

This restriction to the movement phase can be missed in the English version of the rules because the sectioning is somewhat unclear;
luckily we can clarify things a little using the title in the German version of the rules: bewegungsphase: siedlung mit siedlerschiff gründen or Movement phase: Establish a settlement with a settler ship.

as a general rule:   


In the build and trade phase you do things for which you need
resources. 
In the movement phase you do everything for which no
resources are needed (exception: buying additional movement points for wool)