Telling the exact time (geçiyor / var)

To answer the question Saat kaç? (What time is it?) with exact minutes, Turkish treats the minutes as either "passing" the current hour or counting down "to" the next hour.

Minutes Past the Hour (Geçiyor)

To state that it is a certain number of minutes past the hour, use the verb geçiyor (is passing). The hour takes the Accusative case suffix (-(y)ı, -(y)i, -(y)u, -(y)ü), while the minutes are stated as a plain number.

Structure: Saat + [Hour + Accusative] + [Minutes] + geçiyor

It is five past three, enter the studio for the broadcast.
Saat üçü beş geçiyor, yayın için stüdyoya girin.

If the hour ends in a vowel, use the buffer letter -y- before adding the suffix:

It is twenty past two, is the news bulletin starting?
Saat ikiyi yirmi geçiyor, haber bülteni başlıyor mu?

Minutes To the Hour (Var)

To state that it is a certain number of minutes to the next hour, use the word var (there is/exists). Because the minutes are counting down towards the next hour, the hour takes the Dative case suffix (-(y)a, -(y)e).

Structure: Saat + [Next Hour + Dative] + [Minutes] + var

It is six to seven, please stay on this frequency.
Saat yediye altı var, lütfen bu frekansta kalın.
It is eight to nine, we are connecting to the news bulletin.
Saat dokuza sekiz var, haber bültenine bağlanıyoruz.

Numbers ending in hard consonants undergo consonant mutation before vowel suffixes. Notice how dört (four) becomes dörd- in the dative case:

It is seven to four, after the advertisements end, the broadcast schedule continues.
Saat dörde yedi var, reklamlar bittikten sonra akış devam ediyor.

Using Quarters (Çeyrek)

Instead of saying fifteen minutes, you can use the word çeyrek (quarter). It fits directly into the exact same structures without any changes:

The schedule is starting on the new frequency, it is a quarter to five.
Yeni frekansta akış başlıyor, saat beşe çeyrek var.