If you right-click the Color Ramp bar in your ArcMap layer properties you can get to the colours that it is applying (could be more than two, but you can find out their colour values from this box)
You're using a unique value renderer applied to what looks like many hundreds of unique values. There are only so many colours that can be found between the start and end colour values of your colour ramp, so that would explain why they are being repeated on your map.
Nevertheless, in QGIS, you would apply the same ramp by:
- Open layer properties of your polygon layer
- Under the Style node on the left, choose the Categorized renderer
- select the Colour ramp icon on the right
- scroll down and choose "New Colour Ramp"
- Choose "Gradient"
- In this dialogue you can build the same colour scheme as you had in ArcMap (I used the 'pick colour' option to use a colour picker eye dropper to replicate your colours, and came up with this:)
That's the basics of it, but there are no ways to automatically do this, so it will require some manual work to get at the colours, keep a list, and apply them to a QGIS colour scheme.