Is there a way to increase water flow in a Kohler toilet model K3467? Water inlet valve is open fully with normal water pressure. Thank You Brian

Either...
Low water pressure.
The inlet valve isn't opening fully.
Small bore pipework.

Is the water pressure good in the rest of the property?
Is so then there may be an isolation valve on the toilet which isn't fully open.

replace the small rubber washer in the water inlet control valve .

It is really the valve that opens up to let water flow through the bowl when activated by the flush lever, to flush the toilet. it should prohibit water from flowing after the toilet flushes.

Valve fully open on inlet going into toilet if it is then fill valve needs to be replaced check water line for any abstructions that might interfere with water going into tank.

it's the fill valve that squeaks. Sometimes a slight adjust to the water level works, but most of the time it needs to be replaced

The flapper on these toilets is a water saver. The water saver flappers can be replaced with a standard flapper and I have found this will help considerably. The flapper is located in the tank at the center bottom. It is the part that flaps up and down when you pull the lever.

You cannot adjust the pressure on the toilet. But you can adjust the stop setting screw at the top of the float arm. If the water is coming in too fast you can adjust the valve where the toilet water line connects to the wall water supply. Closing the valve a little will slow down the flow of water and it will take a little longer to refill the toilet water tank.

replace the valve rubber at the top of the inlet control

Double check that washers inside fittings attached to back of tank, are not wedged or pinched in there restricting flow. Drain tank, relieve water pressure prior, obviously. The other thing is to check that the by-pass is allowing good flow in to tank when in normal operation mode. Have seen those only go half way inside valve, but lever is at wide open setting. If all that checks out, try removing the check valve on tank, and replace it with a 1/2" galvanized pipe nipple, ( or certified plastic/nylon fitting same size) and reconnect water line direct to that, without a check valve. That should do it if nothing else was wrong. Now you have a choice, either change it or leave it out if everything seems ok, but double check that hot water isn't siphoning back to another utility such as toilet if you leave it out and toilet is close. Feeling that cold line (inlet) when water is hot should tell you. I've seen the check valve in both locations, hot (at top of tank) and cold (at bottom of tank -inlet) Not 100% sure why, right from factory that way, but I say, put it on bottom fitting if required at all. Let me know how you make out.

I had exactly the same problem. Just turn down the water pressure @ the valve on the water inlet line. While you have the toilet lid off, you may want to remove the fill valve (1/2 turn counterclockwise on the plastic nut attached to the water-in line), and clean the rubber diaphrams (there are 2). They will sometimes get gritty and this will cause some fill noises and leaking.