Placement of infinitive

In the first sentence, "to go" appears to be somewhat far away from "expect". I wouldn't use that construction; as I see it, this is a matter of style, not really of incorrect grammar.

The second sentence is not correct grammatically¹.

Here is one construction that seems to be more direct and that has the advantage of not using "between" after "to go", thus eliminating the possible idea that "between" is associated with "to go" rather than "debate" (although this latter case appears readily enough to be the only choice).

• How do you expect Marjorie Taylor Greene and Alexandria Ocasio-Cortez would debate with one another on the green new deal?

In this last formulation above, the idea of how the debate would turn out has been rather slighted, and so the next options could be found to be better.

• How would turn out a debate between Marjorie Taylor Greene and Alexandria Ocasio-Cortez on the green new deal? What do you expect?

• How would turn out a debate on the green new deal between Marjorie Taylor Greene and Alexandria Ocasio-Cortez? What do you expect? (For the uninformed, the deal could be interpreted as being between Marjorie Taylor Greene and Alexandria Ocasio-Cortez, which the writer or speaker might want to avoid.)

¹_{Correction due to user BoldBen}