Remember that h takes an element of Q to an automorphism of K. In this case, acts on the element . And because it is an automorphism, it is multiplicative. Therefore . Also, . Thus
You can't write h(q) on its own here; h(q) is an automorphism, and it has to act on an element of K. In this case, it acts on the identity element of K, and an automorphism always takes the identity to the identity. Therefore h(q)(1) = 1, and (k.h(q)(1),q.1) = (k.1,q.1) = (k,q), as required.
Afterthought: I think you would find this whole calculation easier to follow if you change the notation for an automorphism. Instead of writing , write it as . That will remind you that is an operator, that has to act on an element of K. It satisfies the conditions , , and .