Combine sugar, Greek yogurt, cream cheese, vanilla extract and lemon juice into large bowl and mix on low speed with a hand mixer until sugar dissolves.
Turn on ice cream machine and add yogurt mix. Let mix until thickened, about 20 to 25 minutes.
During last 5 minutes of freezing, add the chopped blueberries and graham cracker pieces. The yogurt will have a soft, creamy texture.
Serve immediately as a soft serve or freeze to an airtight container and place in freezer for several hours for a firmer texture.