Income tax is "pay as you go." You are supposed to pay tax on your income as you earn it during the year. Estimated tax payments should be made as follows. Note that the payment periods are not calendar quarters.
Income earned Jan 1 - Mar 31, pay by Apr 15
Income earned Apr 1 - May 31, pay by Jun 15
Income earned Jun 1 - Aug 31, pay by Sep 15
Income earned Sep 1 - Dec 31, pay by Jan 15 next year
You have to pay a penalty if you do not make timely payments.
Another option, instead of making quarterly estimated tax payments, is to give your employer a new Form W-4 to increase the withholding from your pay to cover the tax on the capital gains. An advantage of doing it this way is that it doesn't matter when the tax is actually withheld, as long as the total withholding for the year is enough. Tax withheld from your pay is always treated as having been paid on time, even if a larger portion of it is withheld late in the year, so you can avoid a penalty.
For all the details on tax payment requirements, see IRS Publication 505, "Tax Withholding and Estimated Tax."
