I know that the Hermitian conjugate of a quantum operator $\hat{Q}$ can be represented as:
$$\displaystyle{\left\langle\phi_1,\hat{Q}\phi_2\right\rangle= \left\langle\hat{Q}^\dagger \phi_1,\phi_2\right\rangle}.$$
In integral form, how would I write this as an integral with $\phi_1$, $\phi_2$, and $\hat{Q}$, as well as the form with $\hat{Q}^\dagger$? Taking the conjugate of a Hermitian is what is confusing me.
Recall that a Hermitian operator is its own conjugate, that is, $Q^{\dagger} = Q$. Then, we have: $\displaystyle \int \phi^{*}_{1}(x) Q \phi_2 (x) dx = \int Q^{\dagger} \phi^*_1(x)\phi_2(x)= \int Q \phi^*_1(x)\phi_2(x)$.
I think you have misunderstood the notation.
When you write $$ \left( Q^{\dagger}\phi_1\right)^* $$ the order of operations matters. First, one has to get image of $\phi_1$, through the operator $Q^{\dagger}$, which is a function in a Hilbert space of complex functions, and then the complex conjugate is obtained by complex conjugation of the function $Q^{\dagger}\phi_1$.
In no place conjugation of an operator is required.
Maybe a possible cause for such misunderstanding could be the reference to the case of a matrix representation of operator. However, in that case there is no integral representation anymore.
