The equation for the volume flow rate is

Q = V/t = Volume/time  .......1

The volume of a portion of the fluid in a pipe can be written as

V = Ad .........2

where A is the cross-section area of the fluid and d is the width of that portion of the fluid. 

Now, from equation 1 and 2, we get

Q = V/t = Ad/t ..........3

Again, the term d/t is just the length of the volume of fluid divided by the time it took the fluid

to flow through its length, which is just the speed of the fluid.

So, we can replace d/t with v in the equation3, we get

Q = Av

where A is the cross-section area of a section of the pipe and v is the speed of the fluid in that section.