If you converted on 6/1 to a rental property, and it asking you this question when you are entering information about the rental, your personal use days are zero. It is asking the question about personal use days once the property was converted to a rental.
I highly recommend that you do NOT have Turbotax figure out how to allocate property tax, mortgage interest, etc. between the personal side (Schedule A) and the rental side (Schedule E). While it can in theory do the calculation, most often people enter in information incorrectly (like putting in a non-zero number for personal use days), or they double count mortgage interest, etc.