Can you cut ridge into fireplace hearth to fit a floor?

Depend weather there is any gas pipes

I think I understand what you’re trying to do…you wish to undercut the hearth (probably by taking 20mm+ off the bottom, or whatever the depth of flooring it is you intend to install). Naturally, you’ll need to ensure there are no other services embedded into the hearth stone (although this is unlikely given where you intend to cut, the previous response is correct in making this point). A careful incision can be made using a thin diamond disc with an angle grinder, but it is likely you’ll need to be confident and competent in its use and operation. Diamond discs are easy to obtain and cut cleanly. The ‘waste’ can be chipped off, but do be careful not to lever the waste off by leveraging against the hearth as you’ll most likely crack and break the hearth piece you’re looking to retain. I hope this makes sense?

Best to get new hearth with up stands set back , made for wooden floors /ect , to fit nicely under without needing edging.

In many cases it is possible to achieve this look however there are a few think to think of, it is common for gas pipes and wires such as sky cables to run around the front of a hearth so it will pay to check if any are in the area, also the appliance that is on the hearth must have a minimum distance to combustible materials. That being said you are unlikely to impede any regulation by simply grooving out to give a neat finish to the floor. The best way to cut that groove would be with a 4” grinder with a grout rake blade to cut the groove, be warned it will be super dusty!