Why does my Honda Accord 2004 kind of grind while turning on? 127k miles?

Whenever I turn on my car it kind of grinds in the way that it kind of like screeches grinds in a way nothing bad with the car though no lights come on or anything.

High pitch grind, could be a bad alternator overunning pulley. Lower pitch grind, could be a bad starter bendix drive. See a mechanic.

The only thing that could Grind when you Crank (turn on) the Starter Drive kicks into the teeth of the Flywheel or Drive Plate. If there's a problem with either pieces it can grind. Sorry to say the only way to check it is to remove it. There may an Inspection Plate you can remove to look at the Ring Gear on the drive plate. Look at it As soon as possible, you DON"T want to damage the Ring Gear, that will be an Expensive job. And every time you hear that grind, something is getting more damaged. You hope and pray that it's the Starter.

Replace the bad starter motor before you chew up the flywheel teeth and have to take the tranny out to replace the flywheel.

There may be an issue with the starter motor clutch. This is part of the starter motor that allows the starter to "slip" when the engine fires and spins up to a thousand rpm. If the starter did not disengage it would be damaged. When the clutch does not release properly you hear a graunching noise, similar to when you accidentally hit the starter when the engine is running.