The answer is no.
In a compensated surrogacy arrangement with a gestational carrier, the baby's DNA comes from the intended mother's egg [or an egg donor] and from the intended father's sperm [or a sperm donor.]
The baby has its own blood flowing through its veins from the DNA of the egg fertilized by the sperm it was created from. The placenta in the surrogate mother provides nourishment to the growing baby and also functions as a filter. The placenta prevents blood as well as DNA from passing between the baby and the surrogate. Recent studies have shown that cells can cross the placenta barrier but there is not sufficient enough information to impact the development of the child.
The baby and the surrogate are not related to each other in any way in a gestational surrogacy arrangement.