How to decide the diameter of a propeller and number of propellers needed in a UAV I am designing?
There is an analytical process for selecting a propeller, but in the most straightforward cases it begins first with the selection of the engine. The propeller choice then can be made almost immediately.
Since you state that the plane will be driven by a turboprop, you have greatly narrowed down the propeller possibilities, and once you have selected a horsepower requirement for your turboprop engine, you have automatically narrowed down the propeller choice even further. Then, once you have selected a particular turboprop engine from a specific manufacturer, your propeller choices are narrowed down to just a couple of possibilities.
Just about the only design choice remaining after you select, for example, a PT-6 with a certain shaft horsepower output specification, will be the prop diameter needed to achieve the correct ground clearance for the engine installation, which may (for example) dictate four blades to absorb the engine power for a small diameter blade disc rather than three blades for a larger diameter disc.
Note also that no prop manufacturer is going to design for you a custom prop from scratch unless you can guarantee the purchase of tens of thousands of propellers of that custom design, and the design process will require years to go from an empty CAD file to a certificated prop being bolted onto an engine.
From an engineering practicality standpoint, your best bet would be to select the engine and then choose the most commonly-used prop in current manufacture with the longest production run and the greatest number of units in the installed base.