First, apologies for stating the obvious if any of this information seems rudimentary to you. But here are the basics as they relate to the burping portion of your question.
Burping can definitely be the result of a change in diet. As the bacteria that make up our gut flora do their job in aiding in digestion, they release gases. Some happen quickly and cause burping, others later in the process after food leaves the stomach causing flatulence. Some bacteria are considered bad, some good, and even yeast plays a role in keeping things in balance.
Studies have shown that changes in gut flora can react to changes in diet in as little as 24 hours. Some bacteria prefer to eat meat or dairy and multiply, others thrive on fruits or vegetables. Many people take probiotics in order to increase the good bacteria in their gut. Others find nutritional sources such as yogurt, but various raw veggies also impart good bacteria into our systems.
If a source of the burping doesn't have an obvious cause just remembering back to what you ate, then a food diary may help narrow it down. I'd expect burping to happen within just a few hours of eating the food that triggers it. But burping in itself is not considered dangerous.
Burping can also be caused by simply swallowing air. Chewing gum, smoking, or drinking carbonated beverages can all make you burp ... and can cause bloating if the air doesn't get released before passing into your intestines.
A diet heavy in carbs can cause bloating if they aren't well digested. This could be a reason why someone may see an improvement in bloating symptoms if they try a gluten-free diet but don't have Celiac.
H. Pylori (a bad bacteria) is also known to cause both burping and bloating along with other symptoms.
Low stomach acid can cause both bloating and burping.
Some sugar alcohols are not well tolerated by some people and may cause burping.
Hormones can affect bloating as well, though I'm guessing you're already aware of cyclical changes, and bloating likely wouldn't be the only symptom if you had a hormone imbalance. But there are some other female issues that can cause you to feel bloated.
And there are other food intolerances other than gluten that may also cause bloating as a symptom.
If I were you, I'd probably start with doing some more research on a variety of topics. You know yourself best so you can weed out some of the possible causes of your bloating or burping, realize that you may also have some other symptoms, or learn a few things to avoid (such as chewing gum or carbonated beverages) or discover that one of your favorite foods is a known culprit for some of your problems.
Based on just symptoms of bloating, I would not presume that Celiac was the cause. And that is a good thing considering how difficult living gluten free can be. However, since you've already started down the road of paying more attention to what you eat, keeping a food/symptom diary is a great way to help discover possible sources of problems.
If the problem were just a matter of the gut flora being out of whack, eating meals that have variety can help, ie. not having meals that are all carbs, all sugars, all meat, etc. And one thing that could help anyone is to research the foods that contain probiotics naturally and add more of them to your diet. Taking a probiotic pill helps some people, but causes stomach upset for others because many of them just contain too much of a good thing.