Can I trust that my Geiger counter (GQ GMC 500) is in fact measuring radiation?

If you are referring to 'radiation' as in heat radiation, then no, your counter won't detect it as it is non-ionizing radiation.

The GMC-500 (AFAIK) is equipped with two J305 or M4011 GM tubes. These will detect gamma, beta and xray ionizing radiation and that is about all. Geiger counters can be susceptible to electric interference, RFI/EMI etc. but I have only seen that in handful of cases over the years, and mostly caused by cheap or faulty power supplies. If your counter is running on battery power and a reasonable distance from sources of interference, then what you will be seeing on the display is actual radiation - either background, or from an actual source.

ChrisLX200 has a good point, and this would be worth testing to see if your counter is affected by UV or not. I have a couple of J305 tubes, one is UV sensitive (older) and the other one (newer) isn't, so not all J305's are equal. 

Your readings look absolutely fine. Text book background radiation. It differs around the world. Some are a higher background, some are lower. I'm in Blackpool, UK and your readings are very similar to my own. Background monitoring does go up and down like that. I haven't seen a background counter that doesn't have graphs like that.

This is one of my stations ( here ) and you will see they are very much the same. This is another of my stations ( here ), and you can see a slight rise in the 7-day graph. I don't know what caused it, but I think I got a source a little close to it for a short time. They are both indoors, in my office. Anyway, my point is, that kind of rise would be something to note with background monitoring. One of Juzzie's stations ( here ) (I'm pretty sure) is in a Radon area, and that can be seen on the graphs with it oscillating up and down as the Radon escapes and then slows, etc etc. It is fluctuations and/or peaks/rises that indicate the presence of  radiation above background level.

I don't think radiation is your issue. Besides the graphs, as far as I am aware, radiation doesn't affect the body such as your symptoms. Radiation is generally not felt. It can't be seen (except in special circumstances), it can't be felt, smelled, heard, nothing. Here is a rundown of how radiation affects the body:

What the radiation does:
Alpha emits 2 protons + 2 neutrons → a helium nucleus. However, alpha can be stopped by a thin piece of paper, so wouldn't affect the body at all, unless it was ingested or inhaled.
Beta+ emits positrons and also emits a neutrino. Can penetrate skin slightly, but worst when ingested or inhaled. Will require thin shielding. 4mm Aluminium will pretty much stop Beta.
Beta- emits electrons and also emits an antineutrino. Same as Beta+
Gamma emits high energy photons. This can penetrate deeply into the body. Depending on the isotope (Uranium, Plutonium, Strontium, Caesium etc) and the actual mass of the source, this can be shielded by a few millimetres of lead, up to several feet thick concrete and metal walls.
X-Rays emit slightly lower energy photons than Gamma. Like Gamma, but lower energy, so the shielding can be thinner.

How it affects the body:
Alpha if inhaled or ingested (e.g., radon progeny) can cause local damage, especially to lung tissue and also causes double-strand breaks in DNA. It can mutate cells or cause cell death, which in turn can lead to cancer.
Beta can cause single and double-strand breaks in DNA. It creates free radicals in body fluids and leads to indirect DNA damage too. High-energy beta sources (like Sr-90 or Y-90) can cause severe burns or bone marrow suppression.
Gamma causes indirect DNA damage via water ionization leading to free radicals. It can travel through tissue, hitting cells and organelles on the way. High doses = radiation sickness, burns, cancer. The higher the dose, the faster it affects the body. Low, chronic doses = mutations, ageing, or long-term cancer risk.

[Ass covering disclaimer] I'm not a doctor, nor scientist, so the following is simply my own opinion. Please seek other opinions from professionals and do not rely on what I say - Radiation affects the body very differently to how you describe your symptoms. Headaches and nausea would only be caused by a high dose, whether in a short or long period, and I think you would be suffering with other things such as organs shutting down, hair loss, marks and blemishes on the skin when you got to that stage. Your readings show a very normal background reading. Even with a high background of ~100 measured on your counter, it may take years, if ever, for it to do harm. And then would be small, but potentially may cause cancer. 

So in a nutshell,
Lower doses over a short period will probably do little harm, and if your immune system is healthy, it will do it's job in getting rid of the damaged cells before they become cancer.
Lower doses over a long period may do harm, but again, your immune system may be able to cope, if you are healthy. It will have more of a chance for damaged cells to become cancer if the body cant get rid of them quickly and effectively enough.
High doses over a short period may cause physical damage, damage cells and risk them becoming cancer, as well as upset internal organs. It is often recoverable with treatment, but may require removal of the affected area/part of the body in order to recover.
High doses over a long period, if undetected, or simply left will do the same as 'high doses over a short' period, but will have more effect, and will most likely become cancer.
Extremely high doses, even for a short time can cause burns, affect body parts and internal organs, to the point they shut down. Major cellular damage and too much for even a healthy immune system to cope with. Symptoms are radiation sickness, headaches, nausea, skin blemishes and marks increasing in severity. Depending where the body is mostly affected it can cause central nerve damage, loss of sight, possibly hearing and smell too. Organs shut down and leads to death.

So with all of that in mind, I don't think your issue is radiation. And I do wish you well, and hope you get over it soon!