Whether the following function is convex or not？

It is enough to show that $h(y,z) = \|y\|^2 / z$ is convex over $\mathbb R^m \times \mathbb R_{++}$, since composition with affine transformations preserves convexity.

We also have $h(y,z) = \sum_i y_i^2 /z$, so it is enough to show that each component function is convex (addition preserves convexity).

Some further thought show that the problem reduces to proving $g(u,z) = u^2 / z$ is convex over $\mathbb R \times \mathbb R_{++}$. The epigraph of this function is $$ \left\{ (u,z,t) : u^2 \le t z,\; z > 0 \right\} = \left\{ (u,z,t) : \begin{pmatrix} t & u \\ u & z \end{pmatrix} \succeq 0,\; z > 0 \right\} $$ which is a convex set, since the PSD cone is convex. This proves the convexity of $g$.