CORRECTION: The Number of Mentions of Moses in תנ''ך is 769; In Torah, 647

This is embarrassing! I erred in my counting the instances of משה in Torah (here). 

The actual count is 649 - 2 = 647 times. 

Even though this 3-letter string appears 649 times in Torah, apparently referring to משה, and 770 times in all of Tanach, two such references in Torah happen to be two false positives:

One is in Devarim 15:2, the other in Shmot 12:4.

"משה" in both these verses does not refer to the name Moshe!

So the total number of times Moshe is mentioned in Tanach is 768.
In Nach itself it appears 768 - 647 = 121 times. 

NOTE:  The Concordance of אברהם אבן שושן says there are 770 iterations of משה. He also mentions both false positives by bringing those two sources, but each in their own context, not as part of the list that comprises Moses' name. 

One that he probably also includes is found in Sefer Shoftim (18:30) where the word מנשה has a superscripted "נ" therein. This name, says , שושן, "refers to משה", as if to exclude the letter Nun from its spelling. As Rashi there explains: "the son of Menasseh. Out of respect for Moses (משֶׁה) a “nun” was inserted thereby changing the name to מְנַשֶּׁה. This letter was written suspended to indicate that it was not Menasseh but actually Moses."

That being the case, my 768 plus this last one, would yield a total of 769 times Moses' name is mentioned in Tanach.

How the Concordance arrived at 770, I still need to discover.

(According to toraware.org, or its app, you can replicate this result by searching for משה with the option: "Ends with these letters". Then repeat this same search, this time for words ending in חמשה. Take this latter's total count and subtract it from the former searche's hit count.  Then subtract two false positives, add 1 because of the name Moses in Menashe, and you get 769.)