Who installed the LQ?  You may try to call them directly.  They are usually very helpfull. 

How does it act when at home?  Is it different than the camp ground?

Are you saying that you did not have it plugged in for two weeks before you used it?  There are meters and gear on the net available to test the voltage at RV sites.  If the voltage was low in the camp ground and your batteries were low when you got there.  The converter may have not allowed you to charge on the supplied shore power.

You may also try at home shutting off your battery switch.  You may not need the batteries in your LQ when on Shore power.  My LQ does not and it saves the constant recharging.

Are you plugged in at home now?  Do you have a 12 volt cigarette type automotive plug in the LQ or access to the batteries?

In other words you need to learn to check your voltage.

No insults intended.  just all the nifty RV things we need to know with an LQ.

A new trailer I bought in 2005 had an electrical problem.

I could plug it into a NON GFI circuit  and the elctrical worked.

If I plugged it in a circuit protected by a GFI it would pop the GFI.

The dealer and the manufacture had no idea of what was causing the problems. I had to wade through all the electrical circuits.  What I finaly found was that there was a Ground and Neutral wire touching on the back of the circuit breaker. So when I was plugged into something that was not protected by a GFI circuit everything in the trailer seemed to be working just fine. But if I plugged into a GFI it immediately pop'd. All the plugs in my garage and outside my home are GFI. So I was getting the imrpession that the trailer was popping all circuits.

Point being, is that sometimes when they are trying to push all those things into a new trailer. they might have bent a wire that is now shorting out or causing some other kind of problem. It may be a defective part that suffered an infant mortality on your first or second outing. So maybe it's not the manufactures fault, just a bad part. Once you find the problem everything will probably be just fine.  LQ just seem to need a few shake out trips to find the flaws and get them under control.

So are just the batteries not getting charged? Do you have the other electrical circuits? Do outlets work? What you are going find is a converter that has 12 volt fuses like you would have in your car, and 110 volt breakers like you would have in your house. If your outlets/AC ect work then the 110 volt side is good, if the batteries are going dead after the weekend then the 12 volt fuses on the charging side may be blown, or the charging circuit (110 volt) of the converter is not working, it may be the wire is not in the breaker, or the breaker is tripped. If you are not familiar with electric circuits then take it to someone who is. Two things to remember about electricity, 1. You can't see it. 2. It will kill you.

Batteries are dead after driving it home on Sunday and circuits would not pull an electric heater on Friday

Are you trying to run a 110 volt heater, off the batteries and inverter????????????

How many hours did you dive pulling the trailer. If you think that by driving down the road will recharge your batteries, then think again. If they are fully charged it may just top them off while the truck is running.... The batteries need to be charged, with a "Battery Charger". If they are way down, weak, discharged, you would probably have to drive to the west coast and back to ever get them fully charged. Is the charger working? Is there a switch between 110 volt and the 12 volt inverter? Are your batteries full of electrolite? Is/does your dealer not know what the problem is? Who replaced the batteries 8 months ago? Is this trailer not under warranty? What make is this trailer? Folks on here ar trying to help you, but there's pieces and things missing from your story. Contact your dealer...

You say.... Batteries are dead after driving it home on Sunday and circuits would not pull an electric heater on Friday

Then you say.... I am not trying to run anything off batteries. There is no electricity to run electric heater and no battery charge at all, even after driving with trailer, which should at least have recharged batteries. Trailer and batteries useless at this point...all I have is gas heat.

I'm a little slow, can't quite figure out what wrong with your rig other that a blown fuse, tripped breaker, switches not in the proper position, dead batteries, etc...

You won't have gas heat without charged batteries because the fan won't run. Trailer being towed by truck will not charge batteries.