GARCH models are applied not directly to exchange rates, but to returns (of exchange rates, stocks, ...). Therefore, you do not need to care whether exchange rates are stationary or not, because they returns are stationary.
In ARMA model, you have to check the stationary of your data than if it is station at level, you can use ARMA. In the case of stationary at level I, you have to use ARIMA
In GARCH model, the stationary data is required to fix model, if data is not, the regression is spurious