Can I Say "Please Count Simon And Me In"?

Yes, you can. "Count me in" by itself is a common phrase meaning "include me." If you want to refer to other people as well, you just put their names before your own - as you have done. You could in theory add as many names as you liked, saying "Count James, Simon and me in" for example.

You might find that some people think this sentence is incorrect and it should be "Count Simon and I in." This is because there is a common error, where people think using "me" in a sentence like this is ungrammatical. It's probably because we are taught very early to say, for example "Simon and I went out" rather than "me and Simon went out." However, in your example sentence it should be "me" because the sentence is accusative (ie, "me" is the object and not the subject.) You only say "I" if "I" is the subject of the sentence.