Well … you could do and yes, it would help to keep the grass green
When you lay turf you want it to grow long roots into the ground and establish. If you lay soaker hose then the roots are very unlikely to grow and therefore the turf won’t establish.
The other thing about laying soaker hose under roots … the roots themselves may wreck it! Plant roots have two main purposes – find food, find water. The roots will find your soaker hose and break their way in! Once you have laid the turf on top, you won’t be able to fix the broken hose!
I wouldn’t put it under turf either. What you could do is have a length of soaker hose that you can just throw on top of the turf when it needs a water. Just remember to take it off once you have finished with it otherwise you will get marks/dead bits on your lawn!