Let \(G_2\) and \(F_4\) be the classical exceptional Lie groups of rank 2 and 4 respectively. In this paper, the authors calculate 2-primary components of homotopy groups of homogeneous spaces \(F_4/G_2\) and \(F_4/\text{Spin}(9)\). More precisely, they determine \(\pi_i(F_4/G_2:2)\) for \(i\leq 45\) and \(\pi_i(F_4/\text{Spin}(9) : 2)\) for \(i < 38\). The main tools are the homotopy exact sequences associated with the 2-local fibration \(S^{15}\to F_4/G_2\to S^{23}\) and with the fibration \(S^7\to \Omega(F_4/\text{Spin}(9))\to\Omega S^{23}\) introduced by D. M. Davis and M. Mahowald [J. Math. Soc. Japan 43, No. 4, 661-672 (1991; Zbl 0736.57020)]. The determination of the group extensions arising from these sequences is done by using Toda brackets, as in Theorem 2.1 of [M. Mimura and H. Toda, J. Math. Kyoto Univ. 3, 217-250 (1964; Zbl 0129.15404)].