The answer is NO. Because identical twins form from a single zygote, the two individuals will have the same DNA.
However, fingerprints are not an entirely genetic characteristic. Fingerprints, along with other physical characteristics, are determined by the interaction of an indivdual's genes and the developmental environment in the uterus.
Genetics determines the general patterns on a fingertip, which appear as arches, loops and whorls. An individual finger can have just one of these patterns or a mixture of them. While a fetus is developing, the ridges along these patterns are influenced by a number of factors, including bone growth, pressures within the womb and contact with amniotic fluid and each baby will have a different uterine experience.
These lead to unique ridge characteristics in each person and finger.
Often, identical twins will have a similar arrangement of patterns, but never the same minute details.