Is there a short form for word Available as in qty for Quantity? I do not even think that "short from" is the right word for what I'm after. If that's so then what is this form called? Shorthand maybe?
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I suggest you "AVBL".
On this website you can find a list of abbreviation.
Edit: Short forms copied from the mentioned website
Business, Real Estate, Real estate jargon
Technology, Telecom, Telecommunications
Business, Property, Property rent
Is there a short form for [the] word Available as in qty for Quantity?
No, there is no established abbreviation for "available" like "qty", however depending on the context of whatever you're writing, you could probably use Av (for Available) and NA (for Not Available - it also means Not Applicable).