Since antiquity astrological tradition have imbued gemstones with otherworldly and mystical properties. Over time astrologers began to juxtapose star signs with gemstones in order to develop their indigenous cosmologies. Among the cultural traditions to adapt star signs to gemstones were the Ancient Indians and Babylonians.
The Western astrological revision which emerged from the Babylonian, then Hellenistic, Zodiac gives us our modern rendition of the horoscope where star signs are paired with gems. For example, Aquarius is twinned with garnet, Pisces with amethyst etc.
As people tend to think of time in terms of the Gregorian calendar more so than the horoscope, a simultaneous tradition of associating gemstones with months has formed. In this respect, one would pair garnet with January as opposed to Aquarius (January 21st-February 19th).
NB. Gemstones are not used interchangeably between the Zodiac and the Gregorian calendar as some gemstones appear in one system but not another.
The custom of wearing birthstones can be traced to fifteenth century Poland and although there have been many worldwide variations of the pairing of birthstones with gemstones, an attempt has been made - due to our modern world being a global-village, no doubt - to standardise the list of birthstones; hence our modern association of November with citrine and December with turquoise.

Zodiac Gemstones
Aquarius = Garnet
Pisces = Amethyst
Aries = Heliotrope
Taurus = Sapphire
Gemini = Agate
Cancer = Emerald
Leo = Onyx
Virgo = Carnelian
Libra = Peridot
Scorpio = Beryl
Sagittarius = Topaz
Capricorn = Ruby
Birth Month Gemstones
January = Garnet
February = Amethyst
March = Aquamarine
April = Diamond
May = Emerald
June = Moonstone or Pearl
July = Ruby
August = Peridot
September = Sapphire
October = Opal or Tourmaline
November = Citrine
December = Turquoise or Topaz