Am I eligible to extend my two-month tourist visa in Thailand for one more month?

Dunno what one you mean by saying the Phuket Airport. there is no immigration office IN the airport to get an extension

There are three immigration offices in Phuket

the Main Office

the Patong Office

the Blue Tree Water Park Office

As Ellie said. Not at the airport.

You’ll need :-

Passport

Fill out form TM7 (available at the office)

Copy passport data page (signed by you)

Copy of entry stamp (signed by you)

Copy of your visa - either the sticker in your passport or pdf evisa (the former signed by you). * if you had one.

Copy of TM6 departure card (signed by you) *if you were issued one - if you entered by air after July ‘22 you wouldn’t have been given one so leave the space on the form blank.

Passport photo (the offices have copy/photo facilities on site or nearby)

1,900 baht

you follow this from the Phuket Immigration Volunteers

you can go when you have 45 days or less left on your current stamp

You can apply for a 30-day tourism extension at one of three offices, Phuket Town (main office), Patong, or Blue Tree. You cannot apply at the airport.

Most offices accept your application when you have 30 days or less with your entry stamp.

Attached is the common requirements list for tourism extension. But better to check with one of the offices for the exact requirements in Phuket or post and ask on Phuket-related groups. (because the TR extension page on the Phuket Immigration volunteer site is outdated)