Could there ever be a rivalry between the angels Gabriel and Michael?

No, there could never be a rivalry between the Angels Gabriel and Michael. Gabriel is known as the messenger Angel, and Michael the warrior Angel. Both of them carry out the will of God Almighty. It is not possible for either one of them to rebel against God, nor is it possible for them to war against each other. They are in the service of the Almighty. They are two of the elect angels.

"I solemnly charge you in the presence of God and of Christ Jesus and of His chosen (elect) angels, to maintain these principles without bias, doing nothing in a spirit of partiality," (1 Tim. 5:21).