If we look at the definitions of "bark" as a verb, we find that the transitive sense of the verb is to say something quickly in a loud voice. We can't "bark people to do something". We can "bark at people to do something" or "bark out orders for people to do something".
So, in the original example, we might write:
"QUIET!" barked the prison guard, trying to force the inmates into silence.
Another way to use "bark" is to "bark a X" or "bark out a X" where "X" is something that can be vocalized, like "laugh", "curse", "order", etc.
Sirius barked a laugh, earning a surprised glanced from the two younger men. (Example sentence from Oxford's definition of bark)
When I looked at the Google NGram starting from 1800 for
* them into silence, I did find "frown them into silence" which seemed like it might be similar to "shout them into silence", but the references were from the 1850s and not the same sort of construction:
...although the leaders of their party in this country ... will be obliged nevertheless to change their plan...from the effervescence which is appearing in all quarters and the desertion of their followers which must frown them into silence. (The Works of Alexander Hamilton)
I included that sentence here just to show that you might find constructions that seem to match "bark them into silence" but they are probably very old-fashioned or not saying exactly the same thing as "x barks y into silence".
Here are the most common words in the modern English corpus that occur before "them into silence" according to Google's NGram viewer.
The "of them into silence" in that list is usually "most of them", "both of them" or something similar.