I would like to have different Qty Increments for different customer groups.
For example retail customers can purchase individual units, but wholesale customers can only purchase in multiples of 6.
Yes, you can by creating a product attribute for the wholesale customer which take the qty multiple for eg: wholesale_qty_multiple. Now save 6 in that wholesale_qty_multiple field for that product.
call this in product view page or were ever qty is called if the customer is wholesale then multiple with this on quantity change if you have plus and minus button.
If you have an input box then create custom validation according to multiples
If you want the same multiple of qty for all products then create a store config field and save value there and use it the same as the product attribute