Technically, it's 2 syllables. That's because generally speaking, "syllable" refers to the actual number of distinct vowel sounds in a word. Vowel sounds are what to count when you're counting syllables, and silent e's on the ends of words don't count as syllables. Sapphire has only 2 distinct/pronounced vowels in it:
Sapph (or Sa) is one syllable, ire (or pphire) is one syllable. Hence, sapphire has 2 syllables.
(Take for example the word "fire". It only has 1 distinct vowel sound, so it's 1 syllable. That same logic applies to the "fire" sound in sapphire.)
You can try looking it up in a dictionary too, most have the phonetic spelling of words, AND divide them into syllables for you too.