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.