The difference I can see (apart from the infrequent occurrence of trimonthly) is that quarterly is associated with splitting the year into quarters whereas trimonthly (or more commonly "every three months") concentrates on the time which elapses between occurrences.
Practically "quarterly" tends to refer to things which affect all users of a service or an entire population on the same date (in your example if the quarterly updates were software updates they would affect all users of the software and would be released to all users on the same dates). On the other hand I and other people have to have vitamin B injections every three months but some people will have them in January, April, July and October whereas I have them in March, June, September and November. The reason for this being that we started having the injections in different months.
Similarly you might pay for something quarterly if the invoice dates are the same for everyone, in fact rents on farms and cottages were usually paid quarterly on the quarter days and, in some cases this still applies. See this website for more information. Modern rental agreements, however, usually start from the date where the tenancy starts regardless of the quarter days. Much of this has to do with modern accountancy practices and systems.
One other difference would be that anything which happens "quarterly" would tend to happen over a period of complete years or, possibly, a lifetime. Things which happen "every three months" are less likely necessarily to occur in full years, contracts could be cancelled at any time for instance. In the specific case of pregnancy the word "trimester" is used to refer to the three-month periods into which the nine-month pregnancy is divided for various monitoring purposes but this is rather different
These differences are not matters of definition but are differences of emphasis and usage. However as others have said "trimonthly" is very rare and is usually expressed as "every three months".