Broken l3 vertebrae after spinal fusion at l4-5 3 years ago is there any relationship?

Yes: Fusion increases stress to vertebral bodies above and below the fusion so fracture risk in those areas is increased. You should also have a bone density study (dexa scan) done if you have not had one in the last 2 years.

Spinal fusion: You can have an injury to the vertebrae above after a fusion but that is very rare unless trauma has occurred in the area. A herniated disc at l3-4 would be more common.