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If $n=2$ the quadratic map $\mathbb{R}^2\to\mathbb{R}^2$ with $(x_1,x_2)\mapsto (x_1^2-x_2^2,2x_1x_2)$ is surjective. This follows because the map $\mathbb{C}\to\mathbb{C}$ with $z\mapsto z^2$ is surjective. Hence there exist real quadratic maps $\mathbb{R}^{2m}\to\mathbb{R}^{2m}$ for all even values of $n$. (Identify $\mathbb{R}^{2m}$ and $\mathbb{C}^m$ and consider $(z_1,\dots,z_m)\mapsto(z_1^2,\dots,z_m^2)$.) For $n=1$ a quadratic map $\mathbb{R}\to\mathbb{R}$ is not surjective. The question can thus be rephrased: for which values of $m\geq1$ is there a real quadratic surjective map $\mathbb{R}^{2m+1}\to\mathbb{R}^{2m+1}$?