No, the first one is grammatical, but the second one is not.
The reason is the repeated to. When you move the interrogative pronoun whom to the front of the question, it turns out to be the object of the preposition to. So moving it splits a phrase. This is OK, and you can just leave the to at the end.
- Whom could we direct our questions to?
However, if you're going to use whom (most people would use who), you probably will want to Pied-pipe the preposition to along with it, to keep the phrase together.
- To whom could we direct our questions?
What you don't do is copy to whom to the front and also keep to at the end.
- *To whom could we direct our questions to?