Can I file taxes? I have a 3-year-old child but I did not earn any income

If you did not receive any taxable income in 2024 then there is no reason to file a 2024 tax return, with or without a dependent child.

If you had no income from working, you are not eligible for any refundable child-related credits like the additional child tax credit.   Filing a tax return will not yield any sort of refund to you.