What species??? We need to know so we can tell you the right temperature, etc.
Once the female has laid the eggs and filled in the hole and returned hungry and thirsty to the branches you can dig the eggs up carefully.
Before you remove the eggs I would have a container ready to incubate them in and a place ready where you can put the container to incubate them.
I use a shoebox sized Tupperware container filled half full of slightly moist coarse grained vermiculite. To test the moisture level take a fist full of the vermiculite and squeeze it. Only a drop or two of water should come out of it. Make small dents in the vermiculite about an inch apart in all directions to set the eggs in. Move the eggs carefully from where they were laid with a plastic spoon without turning them and place them in the dents. Once the eggs are in the container put the lid on the container.
Put the container in a dark spot where the temperature is at the appropriate temperature to incubate the eggs. Check on them once in a while to make sure all is well with the eggs. Don't leave the lid off the container too long when checking.