The purpose of this paper is to determine the ring structure of the graph equivariant cohomology of the GKM graph induced from the even-dimensional complex quadrics. We show that the graph equivariant cohomology is generated by two types of subgraphs in the GKM graph, which are subject to four different types of relations. By utilizing this ring structure, we establish the multiplicative relation for the generators of degree 2n and provide an alternative computation of the ordinary cohomology ring of 4n-dimensional complex quadrics, as previously computed by H. Lai. Additionally, we provide a combinatorial explanation for why the square of the 2n degree generator x vanishes when n is odd and is non-vanishing when n is even.