Composite materials have become extremely important for several engineering applications due to their superior mechanical properties. However, a major challenge in the use of composites is to detect, locate and quantify fatigue induced damage, particularly delamination, by using limited experimental data. The use of guided Lamb wave based health monitoring with embedded sensors has emerged as a potential solution to effectively predict delamination size. To do this, machine learning prediction models have been used in the past, however, a transfer learning approach which can address the problem of inadequate labeled data by allowing the use of a pretrained model for predicting damage in a new composite specimen, has not been explored in this field. This paper proposes a temporal convolutional network (TCN) based transfer learning (TCN-trans) scheme for predicting delamination damage using sensor measurements. The application of proposed framework is demonstrated on Lamb wave sensor dataset collected from fatigue experiments measuring the evolution of damage in carbon fiber reinforced polymer (CFRP) cross-ply laminates. The results show that TCN-trans yields better damage prediction by fine-tuning a pretrained model with a small number of test specimen samples as compared to a TCN trained only on the test specimen data.