Tente isto:
.PHONY: backup
backup:
@mkdir -p ./backup/include #make folder (don't complain if it already exists)
@mkdir -p ./backup/src #make folder (don't complain if it already exists)
@mv -fu *.h~ ./backup/include &> /dev/null #move header file backups into desired folder
@mv -fu *.cpp~ ./backup/src &> /dev/null #move source file backups into desired folder
O &>/dev/null
deve redirecionar toda a saída do comando mv
para /dev/null
, que acaba com isso.