all-acronyms.com states that "Quest" can be used as an abbreviation for "Questionnaire".
I have seen this used before, though the word "quest" has an entirely different meaning then questionnaire does, so I generally avoid trying to abbreviate the word so that there is no confusion.
But in the context that you are using it (in a figure), perhaps if you labeled it as "quest." it would be suitable.
When in doubt, spell it out.
As for general rules, Wikipedia has the following:
If the original word begins with a capital letter, so should the abbreviation. (ex. Volume = Vol.) If the original word begins in lowercase, capitalization is not needed.
The use of periods differs between British English and American English.
In British English, according to Hart's Rules, the general rule
is that abbreviations terminate with a full stop (period), whereas
contractions do not.
Doctor (contraction) = Dr
Professor (abbreviation) = Prof.
The Reverend (contraction or abbreviation) = Revd or Rev.
The Right Honourable (contraction and abbreviation) = Rt Hon.
In American English, the period is usually added if the
abbreviation might otherwise be interpreted as a word, but some
American writers choose not to use one.
To form the plural of an abbreviation, a number, or a capital letter used as a noun, simply add a lowercase s to the end. (ex. Mind your Ps and Qs.)
To indicate the plural of the abbreviation of a unit of measure, the same form is used as in the singular. (ex. 1 min or 20 min.)
Both "w/" and "w/o" were common, very informal, U.S. abbreviations in correspondence, and in tight spots on data tables, until recently. "C/o" has always been used in addressing letters to third parties: "John Jones, c/o Smith family..."
The other "slash/shilling mark" abbreviations were so uncommon that they usually were not used because of lack of clarity.
I personally use only "c/o", "w/" and "w/o" of all the abbreviations shown on this page. (Except for very limited use of the technical jargon abbreviations: I/O, A/C.)
Best Answer
In legal contexts, the abbreviation "v." is used. Elsewhere, the most common is "vs.". In formal contexts (e.g. scientific papers), it is probably best to have the period at the end of the abbreviation. I assume you would be using this abbreviation in graphs/charts/titles and things like that; the abbreviation would be appropriate in these places, but not within normal prose of the paper.