Not necessary.  It depents on the parameters.....
1 ARCH and GARCH are used to model Conditional Heteroscedasticity.
2 Conditional Heteroscedasticity does NOT mean Unconditioanal Heteroscedasticity.
3 Unconditional Heteroscedasticity implies Unstationarity(in the WEAK sense).
4 So conditional heteroscedastic process (ARCH or GARCH) could be stationary, or unstationary, (in the "weak" sense). 
It depents on the parameters of the model.