Can I count a new swimming pool as a major purchases for purposes of counting toward a state sales tax deduction?

Yes, if this was an inground pool, it could be considered a major renovation or substantial addition to your home, and if the sales tax charged was the same rate as the regular sales tax in your state, you could add it to the sales tax table amount to figure your deduction.

If the quote and contract does not show the tax, then do I just figure it at my counties tax rate?

Yes, after you verify that the tax was collected and paid.    

@AjaxDaKid