Just four remarks to these answers: (1) Ling and McAleer (2003) also cover VARMA-GARCH models that are, apparently, not accepted in the rmgarch package correctly suggested by Tihana Škrinjarić; (2) the book chapter mentioned by Tihana Škrinjarić and Kehinde Mary Bello; (3) none of the provided resources mention Ling and McAleer (2003) in their references. Ruey S. Tsay (2014) book on Multivariate Time Series Analysis With R and Financial Applications does this. Ravi Kumar, have a look at the MTS package for R that covers multivariate time series and GARCH models; (4) Also, the SAS statistical software package propose VAR and VARMA plus GARCH models in their PROC VARMAX procedure but I have no experience with it.