How to connect dishwasher waste pipe to sink waste pipe

How much trouble will depend on if you need to keep the quick disconnect device or not. If it needs to remain, you will need to somehow transition from a hose end fitting of the washer connection to whatever threading your disconnect adapter that fits on the faucet uses. I know I've seen faucet to hose adapters, but there will be a gender issue where both adapters are male, requiring a femalexfemale faucet coupler, which I've never seen.

Another issue is how to connect the existing drain outlet to an extension hose running to the washer drain. This may be easy or difficult, depending on the drain outlet configuration. These quick disconnects seem to be prone to slight leakage, not a problem over the sink. Consider the implications of a possible leak in the laundry room.

If you are willing to cut off the quick disconnect (I assume it is like others I've seen that are swagged connections with no way to disassemble it) then it's a pretty straight forward matter of acquiring standard brass garden hose fittings and fitting them to the cut hose ends. Before proceeding with irrevocable cutting, try to ascertain that the inside hose diameters are compatible with available fittings.

As long as the hoses are of compatible size and short lengths of hose remain attached to the quick disconnect, you can reattach the disconnect using brass hose couplers, though of course the couplers will become a permanent part of the assembly from here on out.

All hose fittings must be secured with screw drive clamps for leak proof connections, thus the specification of brass fittings. Plastic fittings will not hold up to the pressures from a firm clamp installation.

The reason for the high loop or air gap in dishwasher drains when installed below the counter are because of the risk of drain water being pushed back into the dishwasher. If you have a clog in the sink and don't have that high loop, water can drain into the dishwasher and flood out your kitchen. So with a countertop dishwasher, I don't see the need for the high loop since the drain hose is already above the level of the countertop.