We live our lives with one eye on the calendar. Yet in RPGs, generally no one but the DM gives a shit about what day of what month it is. The average fantasy world calendar is just as useful as the French revolutionary calendar today.
Here are two ways that will make your players pay attention to the calendar.
Trick # 1 - Blame it on the weather
Just as us modern monkeys live with one eye on the calendar, we live with the other on the Weather Channel. I surmise 99.9% of any RPGers out there have read The Lord of the Rings, and know why Gandalf and crew turned back from the mountain passes and braved the deathtrap Moria - bad weather. Sure it was meddled with, but the fact stands that weather can ruin the most perfect of adventures. Have PCs bake in the summer sun (and run out of water), shiver in autumn winds (until they find cloaks), and bunker down when the snow hits the ground (or else recreate the Donner party). This will teach them that spring is when a person's fancy turns to adventuring, at least until the first flash flood hits.
Trick # 2 - Holiday, celebrate, run for your life!Children especially measure the passage of time by the approach of holidays, and most can tell you exactly how many days till Christmas or their birthday. Here are a few fantasy holidays that are worth having in your world.
AUTUMN - All Souls Eve (Obon or Night of Hungry Ghosts in Oriental campaigns)
On this day the dead return to watch over loved ones or to do unfinished business, the undead can talk, but are peaceful unless the covenant is broken. Now's the time to ask the dead questions...
WINTER - Krampusnacht
Ah Christmas, the night when the Jolly Old Elf visits every household and gives presents to the good and eats the bad. Evil characters better do a good deed to put their name in the right ledger, or else be off world when Krampus makes its rounds, while the good might find just that piece of equipment they needed in their stocking.
SPRING - Carnival
With the return of life to the land the townsfolk come out to dance, drink, eat, and beat the shit out of one another. On this day all hierarchical relations are reversed, wizards must carry swords & abstain from magic, fighters must wear gowns and abstain from violence, thieves must give to the poor and needy, and clergy just leave religion at home, get shitfaced and have a good time. Breaking these customs is a taboo punishable by a knock to the head and time in the stocks.
SUMMER - Festival of Fireballs
To beat the heat, townsfolk gather by a cool river or lake, drink chilled wine, and watch the Wizardworks. Rival magicians from all over converge to flaunt their Fireballs, Mass Illusions, and Prismatic Sprays across the sky for the adulation of the masses who they usually ignore or zap. No one knows why they do this, whether for prestige, to please their patrons, or just because wizards are cray cray.
Considering that D&D is the elephant in the room, here are some extra holidays based on the alignment system you could sprinkle into your game calendar. These can also be great adventure seeds or hints for players.
LAWFUL or GOOD - Day of Feathers
Angels come down to earth and walk amidst mortals on the streets of a holy city. Woe to any but the pure of heart, while those good souls who have done righteous deeds make have questioned answered or ask for divine aid in their next holy mission.
CHAOTIC or EVIL - Purge Night
Just like in the movies or the Rick & Morty episode, the warped townsfolk put on demonic masks and mercilessly slaughter anyone they want for 24 hours. PCs can either join in the fun or wander in innocently and end up trying to stay alive for the duration, with the moral conundrum of whether to slaughter a whole town or not.
NEUTRAL - The Green Time
An animal gives itself for slaughter & feasting to neutral townsfolk, but a child must be slain in return. Strangers are considered as children, so are welcome with open arms, or else must find a sacrificial substitute.
Finally, here are two more random holidays worth fixing the dates of on your calendar.
BLACK DAY OF MERCHANTS
Here it comes again! All overstock must be sold! Beat the crowds and be there to take advantage of these crazy deals in equipment magical and mundane! No theft is allowed anywhere in the vicinity this day, and anyone with sticky fingers will have to deal with the local Thieves' Guild and merchant-owned golem floorwalkers.
THE EXODUS PILGRIMAGE
A usually hostile race passes peacefully through lands of men & women once every x years to commemorate a past exodus. Gillmen join a Hadj to a buried desert city, dark elves sneak into the aboveground city that birthed their race, to light candles or hordes of goblins bathe in the sewers of People Town. Remember that humans and pilgrims honour a pact of non-violence during this time, and woe betide anyone who breaks the sacred immemorial agreement.