Change old external ultegra brake cables to under the tape brakes
Yes you can change your STI brake levers/shifters for ones with cable routing under the bar tape. There is nothing special required.
However I see very little advantage in routing under the bar tape. It looks slightly better (subjectively) and improves aerodynamics slightly. Personally I upgraded from 105 5600 (good old “washing lines”) to 105 5700 due to a broken shifter. I have the impression that the 5700 with cable under the bar tape shifts slightly worse, which wouldn’t be surprising considering the sharp bent required.
I assume these are Ultegra 6600 STIs? They are 10 speed. Unfortunately the only compatible 10 speed shifters with routing under the bar tape are the 105 5700. Ultegra 6700, Ultegra 8000 and 105 5800 are all 11 speed. You’d have to upgrade your rear derailleur, cassette and chain as well.
Edit: Oh, Ultegra 6700 is 10 speed and has routing under the bar tape. Thanks to JoeK for pointing that out. So you could go for either 105 5700 or Ultegra 6700.
The 10 speed Tiagra shifters are incompatible with other 10 speed rear derailleurs.