CORRECTION: The Number of Mentions of Moses in תנ''ך is 768; 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 shows 770 iterations of משה, but I stand by my count nevertheless; More likely the Concordance made the mistake. (Not because of my genius but because the Concordance authors probably got their results in a more cumbersome way, no doubt, because back then search technology wasn't nearly as sophisticated as it is today.)

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 searches hit count. 

Range limits are 1 to 54 for Torah, 1 to 88 for Tanach, or 55 to 88 for Nach alone. You ought to get the counts 649 and 770, for Torah and Tanach, respectively. 

As for the 2 false positives, this knowledge came from learning Torah.