Concerning 'concern'
The government does not concern about the crime rate in cities.
Here concern is used in the form of a verb, but the preposition about does not belong with the verb.
Concern - Adjective
You can (not) be concerned about nothing in particular:
The government is not concerned.
And you can (not) be concerned about a particular thing:
The government is not concerned about the crime rate in cities.
If something is the object of concern, then remember to use the preposition: concerned about something.
Concern - Noun
It's also possible to say the same thing using concern as a noun, where concern often collocates with another word (in this case, 'cause'):
The government does not consider the crime rate in cities to be a cause for concern.
Concern - Verb
And it's also possible to use the verb form (notice the auxiliary verb 'to do' in the construction of the negative!):
The crime rate in cities does not concern the government.
Which form of 'concern' is best?
So which one to use? Adjective? Noun? Verb? Well if you can remember to put __ed onto concern and add about, it's probably best to use the adjective form!
Practice
Tell a friend about some of the things you're concerned about.