Working part time as an independent contractor. I make about $300 per month. Do I need to file estimated quarterly taxes?

Probably not.

A taxpayer is required to pay estimated taxes if the tax owed after all other payments such as payroll withholding is more than $1,000. Your part-time income will not generate tax of more than $1,000, so unless you have other taxable income for which there is not adequate withholding, you are not required to file estimated taxes.

If that's your only income, you don't have to for federal, but be prepared to pay over $5000 when you file. You might or might not have to file quarterly for state or local. If this is a side job for you and you have other income, file quarterly to avoid being penalized.

since your self employment tax is $42.39 a month on the $300, your se tax will not cause you to be required to file 1040ES

if this additional $300/mo for 12 months has enough affect on your AGI, it is possible but you can adjust the withholding on your regular job to make sure it is all covered

If this is your only income, then no you don't have to.

If this extra income and you are already in the 15% tax bracket it will push you over the $1000 cutoff. If you have a regular job, you can bump up the withholding or pay quarterly.

It will add about $500 in self-employment tax to whatever else you owe. That by itself is not enough to require estimated tax payments.