How many bytes are required to store an address of an integer?

32 bits or 4 bytes.

This depends heavily on the processor architecture your computer uses, AND the programming language you are using.

Typically, an integer is 1 Word in length, however that processor architecture defines "Word". That could be 32-bits (4 bytes), 64-bits (8 bytes), 8-bits (1 byte), or even 9 bits (in certain old computers).