Maine (22,890) and Iowa (19,353) both beat out Colorado and California. Why the Maine love I've absolutely no idea.
"Washington" appears 22,246 times, though that's massively confounded by appearances of "Washington Post" (18,634). Adding "state" as a qualifier (e.g., "state of..." or "... state") reduces that to 461 instances. "Seattle" alone appears nearly 5,000 times which would be 9th overall (sorry, Florida). "Redmond" gives another 892 occurrences.
Colorado is actually 11th most frequently occurring state name.
If you include cities and toponyms from within California (SF, SJ, Los Angeles, SD, SV, Hollywood, Palo Alto, Menlo Park, Stanford, etc.), there are well north of 30k mentions of the state or regions within it.
10 most mentioned: ME (22,890), IA (19,353), CA (10,839), NY (9,009, including NYC matches as they're within the state), IN (6,151), TX (5,603), OH (5,085), FL (2,556), VA (2,369, excluding WV).
10 least mentioned: SD (93), RI (98), NH (136), ND (150), NE (159), WV (161), SC (178), WY (209), NM (213), VT (229).
Data (with additional cities/regions or confounding searches listed for some cases):
Alabama: 372
Alaska: 1,127
Arizona: 914
Arkansas: 264
California: 10,839 (Cities/regions: San Francisco: 7,221, San Jose: 493, Los Angeles: 1,540, San Diego: 732, Sacramento: 187, Silicon Valley: 13,782, Hollywood: 3,050, Palo Alto: 833, Menlo Park: 120, Redwood City: 33, Santa Clara: 199, Santa Cruz: 96)
Connecticut: 1,369
Colorado: 1,633
Delaware: 612
Florida: 2,556
Georgia: 1,350
Hawaii: 883
Idaho: 274
Illinois: 789 (Chicago: 3,634)
Indiana: 6,151
Iowa: 19,353
Kansas: 573
Kentucky: 301
Louisiana: 294
Maine: 22,890
Maryland: 921
Massachusetts: 620
Michigan: 1,456
Minnesota: 530
Missouri: 314
Mississippi: 248
Montana: 486
Nebraska: 159
Nevada: 580 (Las Vegas: 582, Reno: 84)
New Hampshire: 136
New Jersey: 491
New Mexico: 213 (Albequerque: 48, Los Alamos: 89)
New York: 9,009 (New York City: 1,905, Manhattan: 2,355, Wall Street: 2,934)
North Carolina: 368
North Dakota: 150
Ohio: 5,085
Oklahoma: 277
Oregon: 911
Pennsylvania: 358
Rhode Island: 98
South Carolina: 178
South Dakota: 93
Tennessee: 287
Texas: 5,603
Utah: 1,005
Vermont: 229
Virginia: 2,530 (less West Virginia: 161)
Washington: 22,246 (though many confounding matches: Washington Post: 18,634, Washington + state: 461, Washington DC: 369, Washington D.C.: 112. Seattle: 4,958, Redmond: 892.) Actual is closer to 461.
West Virginia: 161
Wisconsin: 387
Wyoming: 209