There are three main ways to determine a streams flow rate:

Look it up. Seasonal flow rate data for thousands of streams has been measured by the US Geological Survey and published on the USGS website

Measure it. You can measure flow rate a couple different ways:

For small flow rates you can measure the time required for the stream to fill a 5 gallon bucket. Then take 300 and divide it by the number of seconds it took. This will give you a very accurate flow rate in gallons per minute. (ex. 23 seconds; 300/23 = 13.0 gallons per minute)

Larger flow rates can be measured by installing a weir. A weir is simply a profile with given dimensions that the stream flows through. Once a weir is in place, you simply measure the height of the water. This water height is then put into equations that return very accurate flow rates. A weir is the most reliable and accurate way to measure large flow rates.

Estimate it. Estimating the stream flow may be accurate enough for your application.

Stream flow estimation uses stream bed dimensions(depth and width),water height, edge roughness, and water velocity(measured with at floating object and a stopwatch).