I don't believe there is, or has ever been, any serious rivalry between Star Wars fans and Star Trek fans. At least, not to the extent commonly claimed by mainstream pop culture. I think this purported rivalry was largely made up by people who observe geek culture in much the same way Jane Goodall observes chimps.

Fandom rivalries tend to happen mostly between very similar works, often with one side accusing the other of being a ripoff, of hogging the spotlight of a particular medium/network, etc. Star Trek and Star Wars are not similar enough for this kind of rivalry. Star Trek is sci-fi (both in terms of setting and themes), while Star Wars is fantasy set in space. The former is mostly on TV, the latter is mostly movies. None is considerably "geekier" than the others; while Star Wars is considered more mainstream, it also has a wealth of supplemental material (expanded universe and such) that can answer the needs of even the most obsessive fans.

Plenty of people like both franchises equally, and for different reasons. Plenty of people see them as complementary in wider geek culture and modern Western fiction. Plenty of people prefer one over the other... but I've never seen anyone get into a heated argument over this, or accuse someone of being a "fake geek" for their preferences.

I would argue there is more of a rivalry between the fans of the various Star Trek Series (e.g., Next Gen fans vs. DS9 fans) than there is between Star Wars fans and Star Trek fans.