What is wrong with "Sie werden den Bezirk absichern?" It should also be correct.
Yes, you could also say that. In this context sichern and absichern would mean the same (both have different additional meanings, but "to make something safe" is their basic meaning).
Because of "Akkusativ" case
Bezirk vs Vierteil
It is Viertel. And a "Viertel" is always a city/town quarter while a "Bezirk" is not so well defined and can all kinds of areas.
Any comments to why "Wir werden das Quartier sichern" is incorrect? Is das Quartier only used in Switzerland?
sichern verses schützen
How can I know which one to use?