Let \( G \) be a simple graph with degree sequence \( (d_1, d_2, \ldots, d_{n}) \). The first and second Zagreb indices are defined respectively as the quantities \( M_1(G)=\sum_{i=1}^{n} d_i^2 \) and \( M_2(G) = \sum_{v_i v_j \in E(G)} d_i d_j. \) The Nordhaus-Gaddum-type invariant associated with the second Zagreb index is defined as follows \( \widetilde{M}_2(G) = M_2(G) + M_2(\overline{G}) \), where \( \overline{G} \) is the complement of \( G \), and let \( \mathbb{G}_{n,m} \) be the family of all graphs with \( n \) vertices and \( m \) edges. Let \( \mathbb{P}(G) \) be a parameter of \( G \). A graph \( G \in \mathbb{G}_{n,m} \) is said to be \( \mathbb{P} \)-maximal in \( \mathbb{G}_{n,m} \) if \( \mathbb{P}(G)=\max \{ \mathbb{P}(H): H \in \mathbb{G}_{n,m} \} \). Using an algebraic-combinatorial method, we in this paper show that, for any fixed \( (n,m) \), the extremal graphs in \( \mathbb{G}_{n,m} \) maximizing \( M_1 \) and \( \widetilde{M}_2 \) coincide.