You can trigger this quest at any time during the year, though it requires two things to get started:
  • A Lightning Rod. You need an Iron Bar, a Refined Quartz, and five Bat Wings to create a Lightning Rod. Iron Ores and Quartzes can be collected from the Mines and turned into the necessary Bars via Furnaces; Bat Wings are dropped by Bats, also in the Mines.
  • A lightning storm. Set up your Lightning Rod and wait for the weather to turn foul. Once it does you’ll find a Battery Pack sitting in the Lightning Rod the next day.
Leave your farm, head to the Bus Stop, and walk west of the bus, along the road. Two screens later you’ll be inside a tunnel. Walk along the northern wall and you’ll find an inactive electrical panel. Plug your Battery Pack into this panel and you’ll receive a message. This triggers the quest.

The Mysterious Mr. Qi wants a Rainbow Shell. There are two ways to find them, as far as I’ve seen:
  • The primary method is to search the beach. They wash up there randomly, though I find they show up with in Summer more often than in other seasons. Fixing the eastern bridge (300 Wood) will expand the amount of beach you can search each day.
  • The secondary method, equally unreliable, is to befriend Demetrius. He will occasionally send you items in the mail, and one of them can be a Rainbow Shell.
Take the Rainbow Shell to the Spa / train station, in the north of Stardew Valley. There’s a lonely box on the train platform. Put the Rainbow Shell in the box and you’ll get a note, alerting you to your next task: place ten Beets in Mayor Lewis’ fridge. This may prove a problem, as Beets are only sold at Pierre’s in the Fall. There are two solutions to this:
  • Buy the Beets at Pierre’s and either wait or get the Greenhouse upgrade. You can grow Beets at any time in the year.
  • Head to Calico Desert, once you have that upgrade, and buy the Beet Seeds at Sandy’s Oasis, in the southwest. Again, you’ll need a Greenhouse if you want this to be completely convenient.
Drop the Beets into the fridge as instructed and you’ll get one final task: ‘Give the sand dragon his final meal’. This refers to the skeleton of a dinosaur that’s basking in the Calico Desert, just north of Sandy’s Oasis. You need to feed it a Solar Essence, which can either be collected from the remains of monsters - the lowest levels of the Mines have plenty of monsters that drop ‘em - or purchased from Krobus, the shadowy creature in the Sewer.

Placing the Solar Essence in the beast’s mouth will reveal one last note. Head back to your farm and check the wood pile to the left of your house’s front door. Inside you’ll find a Club Card. This Card will allow you to access the Casino, located in the back of Sandy’s Oasis.