I tried everything I could to make this work, but couldn't ever get a solution within Tableau.  That doesn't mean it isn't possible, but I sure couldn't find a way.

You can achieve this by finding the length of the string then subtracting the length of a string where you've replaced the the characters you're trying to find with nothing and divided by the length of the characters you're trying to find:

(len([Fullname])-len(REPLACE([Fullname],[findstr],"")))

/ len([findstr])

Tom: I am unable to follow your logic--can you provide an example?  Simply copy and paste "John Doe" as a record into Tableau, and apply your solution so we can see it working.  I'm not able to follow what you are suggesting here.

My formula above was the example. Substitute Fullname with the field you're searching, and findstring with the string you want to find.

I've attached a workbook example.

Thanks for clarifying--this is great!  It would be wise to use the UPPER or LOWER value of the string you are searching for, so case sensitivity is not an issue.