Bollobás and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859-865) conjectured that for a graph $G$ with $e(G)$ edges and the clique number $ω(G)$, then $ λ_{1}^{2}+λ_{2}^{2}\leq 2e(G)\left(1-\frac{1}{ω(G)}\right), $ where $λ_{1}$ and $λ_{2}$ are the largest and the second largest eigenvalues of the adjacency matrix of $G$, respectively. In this paper, we prove that for a sequence of random graphs the conjecture holds true with probability tending to one as the number of vertices tends to infinity.