I paid my tuition April 2019 as a F-visa holder with no SSN, and started work since August 2019 under new SSN. Can I include my tuition payment in my tax filing?

It depends on whether you are eligible to file as a U.S resident alien, or must file as a nonresident alien.  Also, if you became a resident alien during 2019, it depends on whether you are eligible under the substantial presence test or because you obtained a green card without meeting the substantial presence test.  

U.S. resident aliens file Form 1040 and are eligible for education deductions.  Nonresident aliens file Form 1040NR and cannot deduct the cost of tuition but may have to report scholarships not used for qualified education expenses as income.

See the section of IRS Publication 519 at this link for more information on the tax rules if you became a resident during 2019.

See also the section of IRS Publication 519 at this link for information on the Substantial Presence test.  Note that your F visa makes you an "Exempt Individual" for a certain period of time, and your days of presence do not count while you are an exempt individual.