what is the meaning of the following sentence-
“he would have an idea.”

Which part are you having difficulty with? There doesn’t seem to be any hidden meaning.
Could you provide more context?

“The commissioner of the Fish and Game Authority would have the public believe that increases in the number of marine fish caught demonstrate that this resource is no longer endangered.”

That example does not contain the phrase, ‘he would have an idea’.
I suspect you are asking about ‘He would have believe’ which is not the same.

he would have the public believe that = he would like the public to think that.

In the writer’s view, the commissioner wants people to believe that because more marine fish are being caught, this is an indication that those fish are breeding in greater numbers and are no longer endangered.