It is in fact two syllables.
Fun fact: Gojira is the combination of 'Go' of 'Gorilla' and 'jira' of 'kujira' (whale). It is said, kujira comes from the combination of old Japanese 'ku' (black) and 'shira' (white) aka 'black white' after the coloration of killer whales. So at some level Jira derives from the word "white". I could probably trace back the roots for shira a bit, but I need to get some work done.
Maybe you're suffering from the usual learning curve thing where non-native speakers have trouble differentiating parts of words, because the emphasis and pronunciation simply isn't what you're used to.
Native speakers don't emphasize every syllable separately, just like you wouldn't emphasize every syllable in na-ti-ve; or how a non-native speaker like me would try to split "separate" into syllables that as se-pa-rate or se-pa-ra-te instead sep-a-rate.
(As someone who knows a couple handfuls of Japanese words, and comes from a language with somewhat similar phonemes, gojira is pronounced exactly how I expect it to be pronounced.)
Aha, okay, that's what I was thinking; I was confused by the phrasing that Wikipedia used (i.e. "the product name comes from the second syllable") and assumed that meant that "jira" was a single syllable (the second), and thought that I might be missing something about how the pronunciation works. It sounds like I was just misreading the Wikipedia quote though, so thanks for bearing with my confusion and explaining it!
Interesting, I would have thought "jira" was two syllables, but I know absolutely nothing about Japanese, so I assume this is correct and the pronunciation rules are just not what I'm used to from English.