Policijska uprava će proveriti zahtev.

Because the adjective has to agree with uprava.

• uprava is a feminine singular noun
• so the adjective also appears in the feminine singular form: policijska

Compare:

• policijski službenik = masculine
• policijska uprava = feminine
• policijsko vozilo = neuter

It is in the nominative singular, because it is the subject of the sentence — the thing doing the action.

So here:

• Policijska uprava = the subject
• će proveriti = will check / will review
• zahtev = the object

It is in the accusative singular, because it is the direct object of the verb proveriti.

The sentence answers the question What will the police administration check?
Answer: zahtev

Because zahtev is a masculine inanimate noun, and in Serbian, many masculine inanimate nouns have:

• nominative singular = accusative singular

So:

• zahtev = nominative
• zahtev = accusative

This is normal. Compare with a masculine animate noun:

• službenik = nominative
• službenika = accusative

This is the Serbian future I tense.

It is formed with:

• a form of hteti as a clitic: ću, ćeš, će, ćemo, ćete, će
• plus the infinitive

So:

• će proveriti = will check / will review

In this sentence:

• će = third person singular future marker
• proveriti = infinitive

Because proveriti is perfective, and Serbian usually uses a perfective verb for a single completed future action.

Here the idea is that the checking will be done as one complete act:

• će proveriti zahtev = will check/review the application and complete that action

If you used proveravati, that would sound more like:

• repeated checking
• ongoing checking
• checking as a process or habit

So proveriti is the natural choice here.

In Serbian, future forms are often written as:

• će + infinitive

That is what you see here:

• će proveriti

You may also see forms like proveriće, but that usually happens when the verb comes first or when there is no word before it:

• Proveriće zahtev.

With a full subject before the verb, Policijska uprava će proveriti zahtev is the most natural neutral order.

Because će is a clitic, and clitics in Serbian usually do not stand in first position in a normal statement.

That is why Serbian prefers:

• Policijska uprava će proveriti zahtev.

not a sentence beginning with bare će.

In a yes/no question, Serbian usually adds da li:

• Da li će Policijska uprava proveriti zahtev?

Yes. Serbian word order is fairly flexible because cases show grammatical roles.

The given sentence is the neutral order:

• Policijska uprava će proveriti zahtev.

Other orders are possible, for example:

• Zahtev će proveriti Policijska uprava.
  This puts more focus on zahtev or contrasts who will review it.
• Policijska uprava će zahtev proveriti.
  Possible, but more marked in style.

So the basic meaning stays similar, but the emphasis changes.

Because Serbian has no articles.

English must choose between:

• the police department
• a police department
• the application
• an application

Serbian does not mark that with separate words. The listener understands it from context.

So Policijska uprava će proveriti zahtev could mean:

• The police department will review the application
• or, in another context, A police department will review an application

Usually the real situation makes the intended meaning clear.

It refers to an institution, not an individual police officer.

Depending on context, it may be translated as:

• police administration
• police department
• police directorate

In official or bureaucratic language, uprava often refers to an administrative body or authority.

It can be either, depending on context.

In official Serbian, zahtev is often used for a formal document submitted to an authority, so in many real-life situations it is best translated as:

• application
• request

It is more formal and bureaucratic than a casual everyday request. So if this sentence comes from official paperwork, application is often a very natural translation.

će is pronounced roughly like cheh, but with a softer sound than English ch.

A few useful points:

• ć is softer than č
• e is pronounced like e in bed, but cleaner and shorter

For English speakers, the hardest part is usually the letter ć. It is not exactly the same as English ch, but cheh is a decent approximation to start with.