If the cracks are small, you could try beeswax.
Rub a piece of solid beeswax against the crack to fill it. Be sure to use pure beeswax with no additives. This wax should be safe, even if the bird chews on the branch. It can also be used as a natural wood polish that is bird-safe, to make cleaning the branch easier.
If the cracks are large, you might try making a simple wood putty by mixing fresh sawdust with white elmer's glue. Apply the mixture into the crack and allow to dry completely. Do this away from your bird, in case of glue fumes. Once cured, elmer's glue should be bird safe. The wood dust will help the glue patch match the surrounding wood.