Is there any graph problem that, when restricted to trees, has $\Omega(n^2)$ time complexity?
Yes. By the time hierarchy theorem, there is a (tractable, solvable in $O(n^2 \log n)$ time) problem $X$ that (on an $n$-bit instance) can not be solved in less than $\Omega(N^2)$ time.

Given an $n$-bit string, I can encode it as a tree, for instance by taking a path $p_1,\ldots, p_n$ of length $n$ and adding an additional leaf vertex to vertex $p_i$ if the corresponding bit $i$ in the string is $1$ (and perhaps adding some constant-size gadget to mark the beginning of the string).

Now I will define problem $Y$ to be to recognize the trees that encode a bitstring that is a (valid) instance of $X$. Clearly this problem requires at least quadratic time (or otherwise we would have a faster algorithm for $X$).

This is of course a rather contrived and artificial example, but there are some  natural problems that are hard for trees as well. See e.g. NP-hard problems on trees.