If you expect to owe at least $1000 at tax time then paying estimated tax during the year is a good idea in order to avoid an underpayment penalty. Take a look at your own withholdings and try the taxcaster tool and the self-employment income tax estimator.
https://www.irs.gov/individuals/tax-withholding-estimator
https://www.irs.gov/pub/irs-pdf/fw4.pdf
Try the Taxcaster tool:
https://turbotax.intuit.com/tax-tools/calculators/taxcaster/
https://ttlc.intuit.com/questions/1901110-do-i-need-to-make-estimated-tax-payments-to-the-irs
https://turbotax.intuit.com/tax-tools/calculators/self-employed/
When you prepare your tax return you will need online Self-Employed or any version of the CD/download so that you can enter your own W-2 income and the self-employment income for which you will have to prepare a Schedule C.
https://ttlc.intuit.com/questions/2926899-how-does-my-side-job-affect-my-taxes
https://ttlc.intuit.com/community/self-employed/help/what-is-the-self-employment-tax/00/25922
https://ttlc.intuit.com/questions/2902389-why-am-i-paying-self-employment-tax
https://ttlc.intuit.com/questions/1901340-where-do-i-enter-schedule-c
https://ttlc.intuit.com/questions/3398950-what-self-employed-expenses-can-i-deduct
Self employed individuals make estimated tax payments quarterly to the IRS because there are no taxes withheld on that income. If you are having additional taxes withheld on your W2 income to offset the self employment and federal income tax he will owe on his return, the only issue would be to make sure you have had enough withheld. If you choose to have your withholding increased, instead of him making quarterly estimates there would be no need for him to submit quarterly payments. The IRS doesn't care where the taxes are withheld as long as you are, in fact, getting them submitted.