Yes. Let $G$ be the closed subgroup of $SO(3)$ generated by those rotations. The complete list of closed subgroups of $SO(3)$ is known. It is
Finite groups: The cyclic group $\mathbb{Z}/n$, the dihedral group $D_{2n}$, the symmetries of the tetrahedron, the symmetries of the cube, and the symmetries of the dodecahedron.
One dimensional subgroups: $\mathbb{R}/\mathbb{Z}$, and $\mathbb{Z}/2 \ltimes \mathbb{R}/\mathbb{Z}$. The former is the group of all rotations about a fixed axis. The latter is the group of all rotations that take that axis to itself, meaning both rotations about that axis and rotations by $180^{\circ}$ about an axis perpendicular to it.
Three dimensional subgroups: All of $SO(3)$.
$G$ can't be finite because it contains an irrational rotation. It can't be one dimensional the angle between your rotations' axes is neither $0$ nor $\pi/2$. So it is all of $SO(3)$, as desired.
EDITED because I left a case off the list before.