I believe the tree you are referring to is a Mulberry tree. The fruit look very much like Blackberries. Unfortunately, they don’t taste as good, at least not to humans. Birds and other animals like them. There is a product called Florel that is for preventing fruit and nut production on trees. It is too late to do this for this year. It is not inexpensive and you have to do it every year because the tree will respond by producing more fruit the following year; it’s a survival mechanism.
As a certified arborist, I have had many people ask about this. I have never done the sprays because of the cost and the long term commitment needed. In the long run you might want to see about cutting the tree down. Since your landlord doesn’t want to do anything about it, you would probably have to pay to have it done yourself. It’s a tough situation and maybe you can work out something with the landlord.