Every resident in Arizona who installs solar panels gets a State Tax Credit of 25% of the total system cost, up to $1,000, to be used toward State income taxes.

The credit is a non-refundable credit. A nonrefundable credit  means that the credit can't be used to increase your tax refund or to create a tax refund when you wouldn't have already had one. The credit can only reduce your tax liability to zero. 

If you can't claim the credit this year, you can carry it forward to future years.

My AZ tax bill for due for 2020 was $44.00 prior to completing Form 310 (AZ Solar Tax Credit).

My installed solar system cost $37000, in 2020.

My AZ state tax credit should be $1000.00.

My tax bill still shows $44.00 due.

Your Form 310 seems unable to compute the tax credit. When I open the actual form on Turbotax, the calculations for the tax credit are missing and can not be amended.

Please explain.

Please contact support via phone for assistance with this; contact details can be found here

Look at lines 51 and 64 to 74 and 76 & 78 ... what amounts do you have on these lines ?