I have blueberries near my apple trees. I’m space constrained so it is tight. my biggest issue is managing my spray schedule for the apples around the blueberries flower and harvest. The timing works but I didn’t think about it ahead of time. If you will spraying the apples you may want to think about the timing of the brambles.

Thanks for the comments!

The pertinent question is how close?

One thing you should be aware of regarding the blackberry bush is that it will spread like crazy. If you should later to decide to get rid of it, it will be a never ending battle.

I'm thinking maybe 10-15 feet away from the trunk. Possibly in raised beds? Raised beds would help prevent the spreading, right? Or would the roots travel under the bed and spread?

The spreading is a separate problem from proximity to the Apple. You'll have to deal with the spreading wherever you plant them. Fifteen feet should be enough imo as long as the tree isn't a great big old fashioned standard. The berry fruit can be managed by regular pruning of fruited canes, strictly training the new canes onto wires and mowing around the bed. I also mulch each side of my raspberry row every year with cardboard and either wood chips or leaves. Every few years I move the row to fresh ground.