I have a long definition for Picard groups, but it is too abstract for me to really understand and apply.
Could someone give an example or two on how to calculate the Picard group for a quadratic integer ring?
Also the relevance of the exact sequence?
If $\mathcal{O}_K$ is the ring of integers of a number field $K$ then ${\rm Pic}(\mathcal{O}_K)$ is just the ideal class group of $\mathcal{O}_K$. For quadratic number fields we have many examples, say, where this group is trivial. In the imaginary quadratic case $\mathbb{Q}(\sqrt{d})$, $d$ squarefree and $d<0$ exactly for $$ d=−1, −2, −3, −7, −11, −19, −43, −67, −163. $$ You will find several computations on MSE of class groups for quadratic number fields. See for example here, how to do it. In fact, a Dedekind domain $A$ is a principal ideal domain if and only if ${\rm Pic}(A)$ is trivial.