Why quaternary nitrogen but not tertiary oxygen?
This has to do with the electronegativity of the species and how good of a base (or how good an acid the conjugate acids) are.

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To put this in perspective, nitric acid ($\ce{HNO3}$) has a $\mathrm{p}K_\mathrm{a}$ of $-1.3$. Yes, $\ce{HNO3}$, a strong acid, is more stable than $\ce{H3O+}$! The central oxygen in $\ce{H3O+}$ is too electronegative to effectively share these electrons and form a stable bond. In $\ce{NH4+}$, its lone pairs are far more available due to nitrogen's lower electronegativity. 

How common each species is only has to do with the number of acids stronger than $\ce{H3O+}$ or $\ce{NH4+}$. As you can imagine, there are going to be a lot more acids stronger than $\ce{NH4+}$, which starts out around the middle of the $\mathrm{p}K_\mathrm{a}$ spectrum, than compared to $\ce{H3O+}$, which is near the end.