In the Book of Numbers Chapter 19, God told Moses and Aaron to tell the Israelites to sacrifice a cow. God told them it has to be a “red heifer”. In the Quran, in Surah 2 Verses 67-71, God is re-telling the story to Mohammad. When God is re-telling the story, He says that it needs to be a “yellow” cow. My question is, is the red heifer from the Book of Numbers Chapter 19 the same cow as the “yellow” cow in the Quran Surah 2 Verses 67-71? Or why are the colors different?

Yes it is the same story but according to quran the last divine book it is yellow and as you have mentioned above it is Red Heifer.
As Quran claimed that
Holy Quran 15:9
‎إِنَّا نَحْنُ نَزَّلْنَا الذِّكْرَ وَإِنَّا لَهُ لَحَافِظُونَ

Indeed, it is We who sent down the Qur’an and indeed, We will be its guardian.
We believe in all holy books and we believe and it is proved it is last message and wherever you see the Quran you will find it is same.Therefore we believe that it is yellow.