NO! I'm telling the truth! Exeggcute are caught in the Safari Zone, or Headbutt Trees!
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safari zone
You'll find a lot of them nothwest of ecruteak city
You can catch and exeggcute in the safari zone, in pretty much every zone. I would go for any lv. 25 that appears. Assuming that you want an Exeggutor, buy a Leaf Stone from the Celadon Dept. Store for 2100 Poke. Use it on Exeggute, and hey presto, there is your exeggutor. Hope this helps.
You can find Exeggcute in the Great Marsh.
Exeggcute evolves into Exeggutor using a leaf stone.