"Coming and going" is an expression that is sometimes used to describe occurrences that are intermittent. It is perfectly idiomatic in British English to use it in the context you are asking about.
On this webpage of the NHS (British National Health Service) you will find the following quotation:
Appendicitis typically starts with a pain in the middle of your tummy (abdomen) that may come and go.
This demonstrates that it is quite common to speak of symptoms "coming and going". "Hiccups" is not only the plural of "hiccup" but also the common name for an attack or spell of hiccuping. As hiccuping is the symptom of hiccups it seems perfectly natural to say that your "hiccups are coming and going".
As it is an expression, it does not necessarily make sense if you break it down. "Hiccups are coming" itself does not sense, but we do say that symptoms have "come on", so you might say something like:
My hiccups came on suddenly
I don't know how anyone could tell that hiccups were "going", but it would be understood if you said:
My hiccups have gone.
Don't confuse the expression with the similar "comings and goings" which is used to describe collected activities.