Hello, I am Dr RB and I am a small animal veterinarian. There are a number of things that cause a choking/choking or gagging sound. The top conditions being hypertension/congestive heart failure, heartworm disease, upper airway disease (such as tracheal collapse or laryngeal paralysis/megaesophagus and lung disease.
The easiest and to rule out is heartworm disease. If Gretchen has had a recent heartworm test and has consistently been taking a monthly heartworm preventative without missing more than 2 monthly doses (in which case the test would need to be repeated). Heartworm disease, while costly, is a treatable disease but some secondary heart changes can occur from it.
Congestive heart disease or systemic hypertension usually presents as intermittent coughing and gagging and exercise intolerance. Sometimes gum color can become pale or even a blue color (e.g. cyanosis) when experiencing the choking. It can be in the presence or absence of a heart murmur or arrhythmia. If Gretchen is still playing, that is good. However, if she tires more easily and this has been a pattern, then a cardiac workup with your veterinarian (including blood pressure, ECG, chest x-rays is probably a good idea.
Upper airway disease can be a little trickier to diagnose. Tracheal collapse is a condition in which the trachea can periodically become blocked when the animal is breathing. It is affects small breed dogs and has a genetic component to it. I can pick up on a lot of tracheal collapse on a physical exam and usually this is easily confirmed on a chest x-ray. Medications can alleviate the symptoms of tracheal collapse pretty well:) Suspicion of megaesophagus can usually be seen by your veterinarian on a chest x-ray and can be concurrent with laryngeal paralysis (part of a rare disease known as myasthenia gravis).
The best advice I can give you is to make an appointment for a physical exam with your veterinarian. Once he/she examines her, he/she will be able to determine what diagnostics are needed first to get to the bottom of her gagging problem.
I hope this was helpful to you in getting you started to determining Gretchen's gagging. If you have any additional questions or need any clarification on anything, please let me know:) If you are satisfied with the response given to you today, please take the time to rate my answer..