So l configure Shimano 600 full gruppo with Shimano 105 dual control shifters.l have 9 gears at the back and the Shifters do l have to change anything on bike is shimano 600 gruppo to be able to run it with the 105 shifters. The shimano 600 gruppo is from around the 90's.Do l have to change the derailleurs will they be able to shift 2x9 gears?

The 600 (Ultegra) gruppo was already 9 speed in the 90 is so there should be no problem as long as the 105 shifters match up as 2x9.

EDIT: Ultegra is still known alternately as 600 to designers, manufacturers, and Shimano. They are one and the same since the mid 80s.

Shimano 600 predates ultegra, the rear derailleur may work with your 9 speed 105 shifters, but it may be designed to work with older 7 or 8 speed chains. If so, you may have to change the jockey wheels, you will need a 9 speed chain, and you may have to buy a new front derailleur if your 600 derailleur does not work with the dual control levers, it may have been designed for down tube shifters.