Tax question- I filed an extension with form 4868. I included 300.00 payment. Where should I note that previous payment in turbotax. Can that payment apply to state tax?

No.  A Federal extension payment cannot be applied to State taxes.

If you use the same exact account you used to file the extension with, you'll see what you paid already included in the Estimates & Other Taxes paid under the Deductions & Credits tab, when you get to that point in the program. If it's not listed there, then you can just add it. But make sure it's not listed there first. Otherwise, even if you unknowingly double-dip, the IRS will still penalize you for the mistake.
If you paid the IRS at the time you filed your extension, then you payment only applies to your federal taxes, not your state taxes. You would have had to file an extension with the state separately, and paid any taxes due the state separately from what you paid the IRS.

If your extension payment did not automatically show up on 1040 line 70 you need to enter it.  So make sure it's not already there before you enter it!

You can type estimates paid in the search box at the top of your return and click Find and it will give you a link to Jump To the screen where you enter Extension Payments.

To enter the amount you paid with an extension (including state extension)  go to
Federal Taxes (or Personal for H&B/Self Employed version)
Deductions and Credits
Then scroll way down to Estimates and Other Taxes Paid
Other Income Taxes - Click the Start or Update button

Next page second section Payments with Extension
Either Visit All or Click Start or Update by the extension you paid