Have a look in the "I want to move to Cozumel" forum.
Mexico only issues actual visas to US and Canadian citizens when they're in the process of obtaining legal residency. They then get a residency status that isn't a visa and the visa stops being valid. Tourists from those and many other countries enter under Migratory Status paperwork as tourists ("FMM"), not with a visa. We can't tell where you're from, so if you're from Lesotho or Chechnya things might be different for you.
There isn't a simple unambiguous answer to your question. Many live on the island full-time with tourist paperwork, which seems work fairly well as long as they leave the country every 180 days or less and re-enter under tourist paperwork. To remain in Mexico for 181 days or longer uninterrupted, you would need either temporary or permanent residency paperwork. Temporary residency is theoretically available for as long as 5 years with a single application but in Quintana Roo in practice it appears that longer than 1 year is not granted. "Permanent" residency is probably for 10 years in reality before a reapplication or renewal would be needed. The current law is less old than that, so that's never happened yet.