In this paper, we consider the problem of ensuring fairness in graph data, more specifically the implementation of fair link prediction. In the context of link prediction in graphs, the fairness notions of demographic parity and equalized odds tries to ensure equal average linking probability and true positive rates across different demographic groups consisting of various node pairs. In this paper, we propose novel algorithms to implement these fairness notions while taking overlapping and intersectional groups into account that arise in the case of multiple sensitive attributes, unlike most of the existing methods which only consider a single sensitive attribute. Assuming that the link prediction model generates a prediction score for each node pair to form an edge, we formulate a convex optimization problem that minimizes the squared Euclidean distance between the original prediction scores and transformed scores, subject to the fairness constraints. The transformed scores are then utilized for fair link prediction. We refer to our overall framework as Fair Link Prediction for Graphs (FairLPG) and demonstrate our method's effectiveness on real-world datasets and graph neural network based link predictions models. It outperforms or performs competitively with existing methods both in terms of fairness and prediction accuracy. To the best of our knowledge, this work is the first to handle the case of intersectional sensitive groups in the graph setting.