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Geometry of conformal minimal two-spheres immersed in $G(2,6;\mathbb{R})$ isstudied in this paper by harmonic maps. We construct a non-homogeneous constantcurved minimal two-sphere in $G(2,6;\mathbb{R})$, and give a classificationtheorem of linearly full conformal minimal immersions of constant curvaturefrom $S^2$ to $G(2,6;\mathbb{R})$, or equivalently, a complex hyperquadric$Q_{4}$, which illustrates minimal two-spheres of constant curvature in $Q_{4}$are in general not congruent.

Rigidity of conformal minimal immersions of constant curvature from $S^2$ to $Q_4$