Preheat oven to 350°F (175°C). Line an 8x8-inch baking pan with parchment paper.
In a medium bowl, combine graham cracker crumbs, 1/4 cup sugar, and melted butter. Press evenly into the prepared pan to form the crust.
In a large bowl, beat cream cheese and 2/3 cup sugar until smooth. Add eggs, one at a time, mixing well after each.
Mix in lemon zest, lemon juice, and vanilla extract until fully combined.
Pour the cream cheese mixture over the crust. Gently press raspberries into the filling.
Bake for 35-40 minutes or until the center is set. Cool completely, then refrigerate for at least 2 hours before slicing into bars.