Definition of ridge regression
$$min_\beta||y-X\beta||_2^2+\lambda||\beta||_2^2, \lambda\ge0$$

you can prove a function is strictly convex if the 2nd derivative is strictly greater than 0 thus

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/j2Ebw.png

But unfortunately I don't know if this is sufficient proof as it's possible for $X^TX$ to be negative and $\lambda$ can be 0. Unless I'm missing something.