Yes you can do that. I use a trellis, not fencing. If you mulch around plants to keep the grass out, but realize that brambles can send suckers 3 feet away from the plant, like in the middle of the grass. Blackberries grow crazy wild. Zone 4 is pretty cold for most types of blackberries. Also blueberries are bushes, not vines and need no support. They do need room, and very acidic soil to grow. Your soil type will matter greatly, and you have to be able to monitor the PH. You need to water with acidic water too. Tap water usually is not good.
I use raised beds for my raspberries so they remain confined to the beds. My blackberries are in ground mulched, surrounded by grass on both sides. here is a photo from one end of my blackberry trellis.

Drew,
Could you elaborate further on the raspberries staying confined to raised beds. That was the first I had heard of that. Northwoodswis

My raised beds are 1 foot tall. the raspberry roots would have to grow one foot down, and underneath. i suppose they could. So far they have not. You could make them taller. I seen some recently that were 2.5 feet high.
I made a 2x24 foot long bed. Other beds are attached, but each bed has a barrier, so to get into the other bed they need to grow up a foot too. If they get into the grass, i don't care, i'll just mow them down. Here is a photo of the bed from june. A side view were you can see construction a little better. The beds to the right are 4x4 and have a blueberry plant in the middle. I can control the soil PH this way. In ground here with blueberries is out of the question. The blueberries are surrounded by strawberries. They will be removed once the blueberry plants are bigger. All plants planted this year. Picture taken on 2013 06 24. The last bed is 8x12 and will be where all the raspberry runners will go, well have gone. it's full now! During the season i grew corn in the bed. i got 65 ears!

Here's a photo of the beds from a different angle on 2013 10 16. Raspberries are trellised with 4 wires.

From a similiar angle as the first photo, but 4 months later, also taken on 2013 10 16, You can see the trellis wires better. Going right is the wires for the blackberries shown in my first post. So bramble patches are at 90 degrees from each other. You are looking Northeast, so this rasied bed runs NE to SW. Blackberry canes can be seen in the forground.

A added 1x4 beds between the 4x4 beds since the photo in June. Next year flowers that attract beneficial insects will be planted in these small beds.
I will prune out a lot of raspberry canes before next spring, Too early right now. Both everbearing and summer bearing types are in the beds.The canes hanging over the 4x4 beds will be cut back or removed. No lack of canes that''s for sure! I could not imagine dealing with them if not confined to the beds!

This post was edited by Drew51 on Tue, Nov 12, 13 at 23:55

Fine looking beds. I bet you are eying the space I see remaining. I'd consider running currants along the top of the chain link fence if you get along with the neighbor and they approve.

"I'd consider running currants along the top of the chain link fence"

Great minds think alike. See the end of the fence farthest away? On that photo you mention? The small part running at a right angle from the main fence? I'm developing a currant cordon on that fence.
Another on the other side. Here is another photo, past my blackberries is my grape vine. First year growth, on the fence, hard to see is the currant. First year growth has it right at the top. Next year I will prune top so it splits to two, and run along the fence top. I will prune ends to keep it on my fence. Currant berries will hang down, it will look awesome!

My dog runs that path, so I have to keep it open. My neighbor would probably object to it, so I used my fence only. The fence really needs to be replaced. Plus I'm moving in 5 years. Yeah sucks. In 5 years everything is going to be so cool! I will have to start over. (sigh)
Sorry I hijacked this thread, my apologies. I hope seeing this gives you some ideas.