There are precious few grills that will, out of the box, work directly off the quick connect propane fitting... The quick connect on your trailer is a low pressure port (the LP has already gone through one regulator) and as such, it doesn't have the pressure necessary to trigger a 2nd regulator that is standard on most portable LP grills.
If you want to use the low pressure quick connect on your trailer, you'll either have to get a grill designed for that use (doesn't have a regulator) or obtain a grill that is easily adapted and good to use. Many in the RV family have chosen the Weber Q series of grills (myself included). It is easily adapted. Robustly built. Heavy components. Top notch in the food preparation (cooking). Personally, I have the Q-120. Used it for years on my 210WBS and am still using it today on my 5er.
If you go that route, the Q series is easily adapted, you'll need a good company that sells the proper fittings. There are many "quick connect" products out there and they are
not all the same. They all look about the same (air, water, LP; low and high, NG; low and high, etc.) but, they aren't designed the same and shouldn't be mixed for safety reasons and most won't actually fit into the wrong socket for that reason. The company I use is Calore Equipment and you'll want the LP low pressure 250 series fittings. They also sell the dust covers, hoses in various lengths; everything you'll need.
Calore Equipment 250 fittings
You'll need to take the regulator off of the grill...it isn't hard to do, just tedious. When you put the new fittings in place, make sure you use yellow Teflon tape rated for gas. That white tape you have floating around is only rated for water and will not do you any good for gas fittings. When you get the regulator off (don't throw it away, I'll show you a use for it momentarily), use the yellow tape and put a male fitting onto the grill in its place. It will look something like this (my grill in the photos):
Yours should look a little neater...I had a bunch of adapters floating around the shop and I used up my "bits", but, the end result is the same, you just won't have most of the piping between the grill and the male fitting. That's really all you need to do to the grill. The hose you have will work with the grill now, but, if you need hoses, the site above has those too:
I carry a selection, a 4' hose, two 10' hoses, and two 12' hoses. I can string all the hoses together if need be, but, I mainly carry them so that I can also use my Campfire-in-a-Can at the same time and be almost anywhere in the campsite. And speaking of being in the campsite...I wanted to go on a picnic off-site at another park (sightseeing kind, not camping kind), but, of course the grill was now tethered to the trailer. Light bulb time...remember that regulator I asked you to hang on to? Take that regulator and get the female quick connect fitting (it is 1/8" though, not the 1/4" that you bought for the grill) and put it onto the regulator.
The top two regulators are for the Q series grill, one is my original, one is a spare that I bought and they fit onto the 1 lb disposable propane bottles. The bottom regulator is actually from my Campfire-in-a-Can and will fit 5 lb and larger tanks with the standard POL fittings. Now I can use my Weber Q around the trailer and use the quick connects, or, I can get a 1 lb disposable and use the quick connect regulator and hose and use my Weber Q off-site. With the bottom regulator, I can use my Weber Q with a spare 20 lb tank at my house for when we have a big party and I need more than one grill...I use my big grill and set up the Q right next to it.
Just another way to go...lots of grills out there. Some are good, some not so much. Some will hook straight up to your quick connect, some need a little adapting, some just plain can't be adapted (those with the gas control and regulator all in one piece).
