You might have supply hoses that have an internal shutoff valve in one end which closes if the flow rate is too large. These are designed as protection against a ruptured hose. One brand is Watts FloodSafe(tm) supply hoses.
If you have these, one might be malfunctioning or connected backwards. This type is unidirectional; the end with the control valve must be connected to the water supply. Standard hoses will work no matter which end is connected to the water supply.
I have a set of Watts FloodSafe hoses in a cabinet, but I have not tried to install them on our new "smart" top loader. I had installed them on the washer of an elderly neighbor, but he was dissatisfied with them because his washer was taking too long to fill. I removed them and put in the standard hoses. It is possible that some of the new washers have sensors to detect low flow. Your washer might be detecting low flow and is programmed to shut off and report an error.
From what I can see the valves MAY BE threaded into a fitting and you can replace the valve with a pair of wrenches. If I am concluding correctly from what I see, the valve itself has male "iron pipe" threads probably (what is called) 1/2" MIP or maybe 3/4" MIP. These threads are also designated NPT.
Alternatively, the fitting below the valve may just be one to allow adjustment to clamp onto the plastic box and it may be that to replace this valve you really do have to cut into the drywall and cut tubing or unsolder a fitting. I think an arrangement like that ought to be a code violation, but I see there are valves sold for that set up. I have our washer valves set up so they can be changed with wrenches and I just assumed that this would be the default modern arrangement, but maybe not. You may have valves like this.
You could cut a hole in the drywall below the box and put in a plastic access port that would allow inspection and verification that joints are water tight.
I would put in a large enough access port to allow the repair to be made. Since this is behind the washing machine there would be no need to restore the drywall to the standard of an exposed wall.
If what I first thought was the arrangement you would measure the diameter at the exposed threads to confirm that it is what is called 1/2" and get a replacement hot water valve with 1/2" MIP threads. These valves are available with either female or male threads so be careful to get the right one. (You could remove the old valve and take it with you, but then water to the entire house (or at least the hot water) will have to be shut off during the time you have no valve in place.
Remove the valve handle if necessary to clear the plastic box when the valve is unscrewed. Put a wrench on the wrenching flats on the fitting below the valve and unscrew the valve with another wrench or with pliers (e.g, channel locks). It is very important to firmly hold the fitting below the valve and not allow it turn as you loosen the valve.
Put pipe dope or teflon tape on the male threads of the valve and thread it in.
Best Answer
In the US you can almost always (almost... there's certainly exceptions) rely on hot being on the left and cold being on the right. This will be the case for washing machine hookups, bath tubs and faucets. I'm not sure if that is a code requirement anywhere, but it is almost universally used. Switching sides would have safety implications because someone would assume the right handle was cold and could be scalded when it ends up being hot.
So...
Also, note that in different regions and even from house to house the valve types can vary quite a bit. The values you have pictured are very modern and sleek and have built in water-hammer arresters. Some houses would just have two gate valves like you would use for an outside water hose with red and blue handles. Sometimes the two valves are linked and only have one handle that turns them both on or off. This Google Image Search shows the wide variety of types out there.