self employment tax question

Whether quarterly payments will be required depends on your entire picture for taxes.  The IRS imposes a penalty if you underpay your total tax liability for the year by more than $1000 - the goal of making estimated payments is to help you avoid that underpayment penalty. 

You mentioned making $650/month as an independent contractor.  You did not mention any business-related expenses that might reduce that income.  For easy math, I'm going to assume you do not have any expenses related to this contracting income.  If you received this amount each month for the entire year, you would have $7800 in self employment income.  You will need to pay both income taxes and self-employment (Social Security and Medicare) taxes on this income. 

The self-employment tax component is the easy part - that is going to be 15.3% of the total income (6.2% for social security, 1.45% for Medicare, and you pay it twice - both the employee part that you normally see on your W2 if you earn wages, and the employer side because when you are self-employed, you are the employer as well as the employee!)  You will get a tax deduction for 50% of what you pay, but it does not result in a dollar for dollar reduction in your tax liability.  

You also will owe income tax on the $7800.  That is going to be based on your tax bracket.  In your case, if you are filing a joint return with your husband and he has $170,000 in wages, you are likely in the 22% tax bracket.  So you can expect that you will pay 22% of tax on your taxable income.  

Keep in mind that if you do have expenses that you incur to help you produce that monthly income, you can deduct those expenses to arrive at your net income from the self-employment, which will reduce your tax liability.    

You probably don't need to go back and try to "catch up" the estimated payments you have already missed for the year.  You can still make the estimated payments due in September and January to help offset any taxes you may owe from your side job.  

Hope this helps!