is it OK to start a sentence with "Pero" ?

"pero" is right. You can also use "aunque."

You wouldn't be starting the sentence with "pero," you'd just be inserting it into the space to make those two sentences one connected sentence. And it would sound perfectly fine.

EDIT: Hmmm. That'd be a rather long sentence. I would start the second sentence with "pero" or use a semicolon, like this:

"Dicen que soy la abeja reina, dando ordenes a ellos, los zanganos; pero yo pienso que ellos no conocen muy bien los bichos."

It is ok to use PERO before beginning the second sentence and after the period. Otherwise, your sentece would be too long. Using a semicolon is another option, and it was very common to join sentences with semicolons. There is even a rule of punctuation that says that before PERO you have to use a comma, or a semicolon if the sentence is too long.

But in modern style, it is better to use a period instead of a semicolon.

its okay to use pero here, because its a continuation of the first part, not a separate sentence. This is all just one sentence.

Just as in English, it´s ok if you do, but better if you don´t.