In my experience, berries can be ripe anytime in August. In a year with a late snow melt (like this year appears like it is going to be) it may take a lot longer. Last year with the fairly mild summer, most blueberries in my area didn't get ripe until the last weekend in August. But I did find one patch on opening weekend that we getting the right amount of sunlight to be ripe, and there was bear sign everywhere.
A lot of experienced bear hunters say that when the berries are not getting ripe, it is the best hunting. Because when you find a small, ripe patch, you will find the bears because they will congregate there. That hasn't always been my experience. Last year in one of my normal spots we found a couple of small, ripe berry patches in mid-August and the bears hadn't touched it yet. So you just never know.
I usually start checking for ripe berries in the last week in July. July would be a very early time to find blueberries, but it is getting close to season. If you find a ripe patch, hunt there opening day, because there is a good chance there are bears there. I have one honey hole where I spot bear about 75% of the time I go there. It is a good spot because it is some nice open meadows (which are relatively easy to reach) where fairly few people hike since the hike doesn't really have much of a view of anything in particular other than the meadows. Since the trail doesn't see a lot of traffic, I think the bear like to hang around in there. I've noticed that they start moving up into the meadows in late July-early August to look for blueberries. Last year, I saw the same bear twice in a week come out into the meadow and walk quickly from one end to the other sniffing around and looking for berries. When he didn't find any, he went right back to where he came from. I think they've learned to look for berries there and they aren't very pressured so they just continue looking until they get ripe.
Overall, I would start looking in late July if you get a chance. If the area is pretty secluded and you know the bear feed there when the berries are ripe, there is a good chance they will also be checking out the area. If you are lucky enough that it is one of the first patches to get ripe, then hopefully you will be able to find a bear early in the season.