Does the US Bureau of Economic Analysis use the generic growth formula to calculate percent change of GDP?
Normally growth is calculated either as:

$$\frac{y_t-y_{t-1}}{y_t}$$

Or approximated by:

$$\ln(y_t)-\ln(y_{t-1})$$

Within year you can either continue using the  above definition or use year on year growth which compares every month quarter with the same quarter previous year.

If you are looking for Fred specifically here is the list of formulas:

https://fredhelp.stlouisfed.org/fred/data/understanding-the-data/formulas-calculate-growth-rates/

The list overlaps with what I written above and also the list shows other changes (which you might be interested to look into)